src/HOL/Probability/Lebesgue_Integration.thy
 author hoelzl Fri Nov 02 14:23:40 2012 +0100 (2012-11-02) changeset 50002 ce0d316b5b44 parent 50001 382bd3173584 child 50003 8c213922ed49 permissions -rw-r--r--
add measurability prover; add support for Borel sets
1 (*  Title:      HOL/Probability/Lebesgue_Integration.thy
2     Author:     Johannes Hölzl, TU München
3     Author:     Armin Heller, TU München
4 *)
6 header {*Lebesgue Integration*}
8 theory Lebesgue_Integration
9   imports Measure_Space Borel_Space
10 begin
12 lemma ereal_minus_eq_PInfty_iff:
13   fixes x y :: ereal shows "x - y = \<infinity> \<longleftrightarrow> y = -\<infinity> \<or> x = \<infinity>"
14   by (cases x y rule: ereal2_cases) simp_all
16 lemma real_ereal_1[simp]: "real (1::ereal) = 1"
17   unfolding one_ereal_def by simp
19 lemma ereal_indicator_pos[simp,intro]: "0 \<le> (indicator A x ::ereal)"
20   unfolding indicator_def by auto
22 lemma tendsto_real_max:
23   fixes x y :: real
24   assumes "(X ---> x) net"
25   assumes "(Y ---> y) net"
26   shows "((\<lambda>x. max (X x) (Y x)) ---> max x y) net"
27 proof -
28   have *: "\<And>x y :: real. max x y = y + ((x - y) + norm (x - y)) / 2"
29     by (auto split: split_max simp: field_simps)
30   show ?thesis
31     unfolding *
32     by (intro tendsto_add assms tendsto_divide tendsto_norm tendsto_diff) auto
33 qed
35 lemma measurable_sets2[intro]:
36   assumes "f \<in> measurable M M'" "g \<in> measurable M M''"
37   and "A \<in> sets M'" "B \<in> sets M''"
38   shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
39 proof -
40   have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
41     by auto
42   then show ?thesis using assms by (auto intro: measurable_sets)
43 qed
45 lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
46 proof
47   assume "\<forall>n. f n \<le> f (Suc n)" then show "incseq f" by (auto intro!: incseq_SucI)
48 qed (auto simp: incseq_def)
50 section "Simple function"
52 text {*
54 Our simple functions are not restricted to positive real numbers. Instead
55 they are just functions with a finite range and are measurable when singleton
56 sets are measurable.
58 *}
60 definition "simple_function M g \<longleftrightarrow>
61     finite (g ` space M) \<and>
62     (\<forall>x \<in> g ` space M. g -` {x} \<inter> space M \<in> sets M)"
64 lemma simple_functionD:
65   assumes "simple_function M g"
66   shows "finite (g ` space M)" and "g -` X \<inter> space M \<in> sets M"
67 proof -
68   show "finite (g ` space M)"
69     using assms unfolding simple_function_def by auto
70   have "g -` X \<inter> space M = g -` (X \<inter> g`space M) \<inter> space M" by auto
71   also have "\<dots> = (\<Union>x\<in>X \<inter> g`space M. g-`{x} \<inter> space M)" by auto
72   finally show "g -` X \<inter> space M \<in> sets M" using assms
73     by (auto simp del: UN_simps simp: simple_function_def)
74 qed
76 lemma simple_function_measurable2[intro]:
77   assumes "simple_function M f" "simple_function M g"
78   shows "f -` A \<inter> g -` B \<inter> space M \<in> sets M"
79 proof -
80   have "f -` A \<inter> g -` B \<inter> space M = (f -` A \<inter> space M) \<inter> (g -` B \<inter> space M)"
81     by auto
82   then show ?thesis using assms[THEN simple_functionD(2)] by auto
83 qed
85 lemma simple_function_indicator_representation:
86   fixes f ::"'a \<Rightarrow> ereal"
87   assumes f: "simple_function M f" and x: "x \<in> space M"
88   shows "f x = (\<Sum>y \<in> f ` space M. y * indicator (f -` {y} \<inter> space M) x)"
89   (is "?l = ?r")
90 proof -
91   have "?r = (\<Sum>y \<in> f ` space M.
92     (if y = f x then y * indicator (f -` {y} \<inter> space M) x else 0))"
93     by (auto intro!: setsum_cong2)
94   also have "... =  f x *  indicator (f -` {f x} \<inter> space M) x"
95     using assms by (auto dest: simple_functionD simp: setsum_delta)
96   also have "... = f x" using x by (auto simp: indicator_def)
97   finally show ?thesis by auto
98 qed
100 lemma simple_function_notspace:
101   "simple_function M (\<lambda>x. h x * indicator (- space M) x::ereal)" (is "simple_function M ?h")
102 proof -
103   have "?h ` space M \<subseteq> {0}" unfolding indicator_def by auto
104   hence [simp, intro]: "finite (?h ` space M)" by (auto intro: finite_subset)
105   have "?h -` {0} \<inter> space M = space M" by auto
106   thus ?thesis unfolding simple_function_def by auto
107 qed
109 lemma simple_function_cong:
110   assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
111   shows "simple_function M f \<longleftrightarrow> simple_function M g"
112 proof -
113   have "f ` space M = g ` space M"
114     "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
115     using assms by (auto intro!: image_eqI)
116   thus ?thesis unfolding simple_function_def using assms by simp
117 qed
119 lemma simple_function_cong_algebra:
120   assumes "sets N = sets M" "space N = space M"
121   shows "simple_function M f \<longleftrightarrow> simple_function N f"
122   unfolding simple_function_def assms ..
124 lemma borel_measurable_simple_function:
125   assumes "simple_function M f"
126   shows "f \<in> borel_measurable M"
127 proof (rule borel_measurableI)
128   fix S
129   let ?I = "f ` (f -` S \<inter> space M)"
130   have *: "(\<Union>x\<in>?I. f -` {x} \<inter> space M) = f -` S \<inter> space M" (is "?U = _") by auto
131   have "finite ?I"
132     using assms unfolding simple_function_def
133     using finite_subset[of "f ` (f -` S \<inter> space M)" "f ` space M"] by auto
134   hence "?U \<in> sets M"
135     apply (rule finite_UN)
136     using assms unfolding simple_function_def by auto
137   thus "f -` S \<inter> space M \<in> sets M" unfolding * .
138 qed
140 lemma simple_function_borel_measurable:
141   fixes f :: "'a \<Rightarrow> 'x::{t2_space}"
142   assumes "f \<in> borel_measurable M" and "finite (f ` space M)"
143   shows "simple_function M f"
144   using assms unfolding simple_function_def
145   by (auto intro: borel_measurable_vimage)
147 lemma simple_function_eq_borel_measurable:
148   fixes f :: "'a \<Rightarrow> ereal"
149   shows "simple_function M f \<longleftrightarrow> finite (f`space M) \<and> f \<in> borel_measurable M"
150   using simple_function_borel_measurable[of f] borel_measurable_simple_function[of M f]
151   by (fastforce simp: simple_function_def)
153 lemma simple_function_const[intro, simp]:
154   "simple_function M (\<lambda>x. c)"
155   by (auto intro: finite_subset simp: simple_function_def)
156 lemma simple_function_compose[intro, simp]:
157   assumes "simple_function M f"
158   shows "simple_function M (g \<circ> f)"
159   unfolding simple_function_def
160 proof safe
161   show "finite ((g \<circ> f) ` space M)"
162     using assms unfolding simple_function_def by (auto simp: image_compose)
163 next
164   fix x assume "x \<in> space M"
165   let ?G = "g -` {g (f x)} \<inter> (f`space M)"
166   have *: "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M =
167     (\<Union>x\<in>?G. f -` {x} \<inter> space M)" by auto
168   show "(g \<circ> f) -` {(g \<circ> f) x} \<inter> space M \<in> sets M"
169     using assms unfolding simple_function_def *
170     by (rule_tac finite_UN) auto
171 qed
173 lemma simple_function_indicator[intro, simp]:
174   assumes "A \<in> sets M"
175   shows "simple_function M (indicator A)"
176 proof -
177   have "indicator A ` space M \<subseteq> {0, 1}" (is "?S \<subseteq> _")
178     by (auto simp: indicator_def)
179   hence "finite ?S" by (rule finite_subset) simp
180   moreover have "- A \<inter> space M = space M - A" by auto
181   ultimately show ?thesis unfolding simple_function_def
182     using assms by (auto simp: indicator_def [abs_def])
183 qed
185 lemma simple_function_Pair[intro, simp]:
186   assumes "simple_function M f"
187   assumes "simple_function M g"
188   shows "simple_function M (\<lambda>x. (f x, g x))" (is "simple_function M ?p")
189   unfolding simple_function_def
190 proof safe
191   show "finite (?p ` space M)"
192     using assms unfolding simple_function_def
193     by (rule_tac finite_subset[of _ "f`space M \<times> g`space M"]) auto
194 next
195   fix x assume "x \<in> space M"
196   have "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M =
197       (f -` {f x} \<inter> space M) \<inter> (g -` {g x} \<inter> space M)"
198     by auto
199   with `x \<in> space M` show "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M \<in> sets M"
200     using assms unfolding simple_function_def by auto
201 qed
203 lemma simple_function_compose1:
204   assumes "simple_function M f"
205   shows "simple_function M (\<lambda>x. g (f x))"
206   using simple_function_compose[OF assms, of g]
207   by (simp add: comp_def)
209 lemma simple_function_compose2:
210   assumes "simple_function M f" and "simple_function M g"
211   shows "simple_function M (\<lambda>x. h (f x) (g x))"
212 proof -
213   have "simple_function M ((\<lambda>(x, y). h x y) \<circ> (\<lambda>x. (f x, g x)))"
214     using assms by auto
215   thus ?thesis by (simp_all add: comp_def)
216 qed
218 lemmas simple_function_add[intro, simp] = simple_function_compose2[where h="op +"]
219   and simple_function_diff[intro, simp] = simple_function_compose2[where h="op -"]
220   and simple_function_uminus[intro, simp] = simple_function_compose[where g="uminus"]
221   and simple_function_mult[intro, simp] = simple_function_compose2[where h="op *"]
222   and simple_function_div[intro, simp] = simple_function_compose2[where h="op /"]
223   and simple_function_inverse[intro, simp] = simple_function_compose[where g="inverse"]
224   and simple_function_max[intro, simp] = simple_function_compose2[where h=max]
226 lemma simple_function_setsum[intro, simp]:
227   assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
228   shows "simple_function M (\<lambda>x. \<Sum>i\<in>P. f i x)"
229 proof cases
230   assume "finite P" from this assms show ?thesis by induct auto
231 qed auto
233 lemma
234   fixes f g :: "'a \<Rightarrow> real" assumes sf: "simple_function M f"
235   shows simple_function_ereal[intro, simp]: "simple_function M (\<lambda>x. ereal (f x))"
236   by (auto intro!: simple_function_compose1[OF sf])
238 lemma
239   fixes f g :: "'a \<Rightarrow> nat" assumes sf: "simple_function M f"
240   shows simple_function_real_of_nat[intro, simp]: "simple_function M (\<lambda>x. real (f x))"
241   by (auto intro!: simple_function_compose1[OF sf])
243 lemma borel_measurable_implies_simple_function_sequence:
244   fixes u :: "'a \<Rightarrow> ereal"
245   assumes u: "u \<in> borel_measurable M"
246   shows "\<exists>f. incseq f \<and> (\<forall>i. \<infinity> \<notin> range (f i) \<and> simple_function M (f i)) \<and>
247              (\<forall>x. (SUP i. f i x) = max 0 (u x)) \<and> (\<forall>i x. 0 \<le> f i x)"
248 proof -
249   def f \<equiv> "\<lambda>x i. if real i \<le> u x then i * 2 ^ i else natfloor (real (u x) * 2 ^ i)"
250   { fix x j have "f x j \<le> j * 2 ^ j" unfolding f_def
251     proof (split split_if, intro conjI impI)
252       assume "\<not> real j \<le> u x"
253       then have "natfloor (real (u x) * 2 ^ j) \<le> natfloor (j * 2 ^ j)"
254          by (cases "u x") (auto intro!: natfloor_mono simp: mult_nonneg_nonneg)
255       moreover have "real (natfloor (j * 2 ^ j)) \<le> j * 2^j"
256         by (intro real_natfloor_le) (auto simp: mult_nonneg_nonneg)
257       ultimately show "natfloor (real (u x) * 2 ^ j) \<le> j * 2 ^ j"
258         unfolding real_of_nat_le_iff by auto
259     qed auto }
260   note f_upper = this
262   have real_f:
263     "\<And>i x. real (f x i) = (if real i \<le> u x then i * 2 ^ i else real (natfloor (real (u x) * 2 ^ i)))"
264     unfolding f_def by auto
266   let ?g = "\<lambda>j x. real (f x j) / 2^j :: ereal"
267   show ?thesis
268   proof (intro exI[of _ ?g] conjI allI ballI)
269     fix i
270     have "simple_function M (\<lambda>x. real (f x i))"
271     proof (intro simple_function_borel_measurable)
272       show "(\<lambda>x. real (f x i)) \<in> borel_measurable M"
273         using u by (auto intro!: measurable_If simp: real_f)
274       have "(\<lambda>x. real (f x i))`space M \<subseteq> real`{..i*2^i}"
275         using f_upper[of _ i] by auto
276       then show "finite ((\<lambda>x. real (f x i))`space M)"
277         by (rule finite_subset) auto
278     qed
279     then show "simple_function M (?g i)"
280       by (auto intro: simple_function_ereal simple_function_div)
281   next
282     show "incseq ?g"
283     proof (intro incseq_ereal incseq_SucI le_funI)
284       fix x and i :: nat
285       have "f x i * 2 \<le> f x (Suc i)" unfolding f_def
286       proof ((split split_if)+, intro conjI impI)
287         assume "ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x"
288         then show "i * 2 ^ i * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)"
289           by (cases "u x") (auto intro!: le_natfloor)
290       next
291         assume "\<not> ereal (real i) \<le> u x" "ereal (real (Suc i)) \<le> u x"
292         then show "natfloor (real (u x) * 2 ^ i) * 2 \<le> Suc i * 2 ^ Suc i"
293           by (cases "u x") auto
294       next
295         assume "\<not> ereal (real i) \<le> u x" "\<not> ereal (real (Suc i)) \<le> u x"
296         have "natfloor (real (u x) * 2 ^ i) * 2 = natfloor (real (u x) * 2 ^ i) * natfloor 2"
297           by simp
298         also have "\<dots> \<le> natfloor (real (u x) * 2 ^ i * 2)"
299         proof cases
300           assume "0 \<le> u x" then show ?thesis
301             by (intro le_mult_natfloor)
302         next
303           assume "\<not> 0 \<le> u x" then show ?thesis
304             by (cases "u x") (auto simp: natfloor_neg mult_nonpos_nonneg)
305         qed
306         also have "\<dots> = natfloor (real (u x) * 2 ^ Suc i)"
307           by (simp add: ac_simps)
308         finally show "natfloor (real (u x) * 2 ^ i) * 2 \<le> natfloor (real (u x) * 2 ^ Suc i)" .
309       qed simp
310       then show "?g i x \<le> ?g (Suc i) x"
311         by (auto simp: field_simps)
312     qed
313   next
314     fix x show "(SUP i. ?g i x) = max 0 (u x)"
315     proof (rule ereal_SUPI)
316       fix i show "?g i x \<le> max 0 (u x)" unfolding max_def f_def
317         by (cases "u x") (auto simp: field_simps real_natfloor_le natfloor_neg
318                                      mult_nonpos_nonneg mult_nonneg_nonneg)
319     next
320       fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> ?g i x \<le> y"
321       have "\<And>i. 0 \<le> ?g i x" by (auto simp: divide_nonneg_pos)
322       from order_trans[OF this *] have "0 \<le> y" by simp
323       show "max 0 (u x) \<le> y"
324       proof (cases y)
325         case (real r)
326         with * have *: "\<And>i. f x i \<le> r * 2^i" by (auto simp: divide_le_eq)
327         from reals_Archimedean2[of r] * have "u x \<noteq> \<infinity>" by (auto simp: f_def) (metis less_le_not_le)
328         then have "\<exists>p. max 0 (u x) = ereal p \<and> 0 \<le> p" by (cases "u x") (auto simp: max_def)
329         then guess p .. note ux = this
330         obtain m :: nat where m: "p < real m" using reals_Archimedean2 ..
331         have "p \<le> r"
332         proof (rule ccontr)
333           assume "\<not> p \<le> r"
334           with LIMSEQ_inverse_realpow_zero[unfolded LIMSEQ_iff, rule_format, of 2 "p - r"]
335           obtain N where "\<forall>n\<ge>N. r * 2^n < p * 2^n - 1" by (auto simp: inverse_eq_divide field_simps)
336           then have "r * 2^max N m < p * 2^max N m - 1" by simp
337           moreover
338           have "real (natfloor (p * 2 ^ max N m)) \<le> r * 2 ^ max N m"
339             using *[of "max N m"] m unfolding real_f using ux
340             by (cases "0 \<le> u x") (simp_all add: max_def mult_nonneg_nonneg split: split_if_asm)
341           then have "p * 2 ^ max N m - 1 < r * 2 ^ max N m"
342             by (metis real_natfloor_gt_diff_one less_le_trans)
343           ultimately show False by auto
344         qed
345         then show "max 0 (u x) \<le> y" using real ux by simp
346       qed (insert `0 \<le> y`, auto)
347     qed
348   qed (auto simp: divide_nonneg_pos)
349 qed
351 lemma borel_measurable_implies_simple_function_sequence':
352   fixes u :: "'a \<Rightarrow> ereal"
353   assumes u: "u \<in> borel_measurable M"
354   obtains f where "\<And>i. simple_function M (f i)" "incseq f" "\<And>i. \<infinity> \<notin> range (f i)"
355     "\<And>x. (SUP i. f i x) = max 0 (u x)" "\<And>i x. 0 \<le> f i x"
356   using borel_measurable_implies_simple_function_sequence[OF u] by auto
358 lemma simple_function_induct[consumes 1, case_names cong set mult add, induct set: simple_function]:
359   fixes u :: "'a \<Rightarrow> ereal"
360   assumes u: "simple_function M u"
361   assumes cong: "\<And>f g. simple_function M f \<Longrightarrow> simple_function M g \<Longrightarrow> (AE x in M. f x = g x) \<Longrightarrow> P f \<Longrightarrow> P g"
362   assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
363   assumes mult: "\<And>u c. P u \<Longrightarrow> P (\<lambda>x. c * u x)"
364   assumes add: "\<And>u v. P u \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
365   shows "P u"
366 proof (rule cong)
367   from AE_space show "AE x in M. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x"
368   proof eventually_elim
369     fix x assume x: "x \<in> space M"
370     from simple_function_indicator_representation[OF u x]
371     show "(\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x" ..
372   qed
373 next
374   from u have "finite (u ` space M)"
375     unfolding simple_function_def by auto
376   then show "P (\<lambda>x. \<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x)"
377   proof induct
378     case empty show ?case
379       using set[of "{}"] by (simp add: indicator_def[abs_def])
380   qed (auto intro!: add mult set simple_functionD u)
381 next
382   show "simple_function M (\<lambda>x. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x))"
383     apply (subst simple_function_cong)
384     apply (rule simple_function_indicator_representation[symmetric])
385     apply (auto intro: u)
386     done
387 qed fact
389 lemma simple_function_induct_nn[consumes 2, case_names cong set mult add]:
390   fixes u :: "'a \<Rightarrow> ereal"
391   assumes u: "simple_function M u" and nn: "\<And>x. 0 \<le> u x"
392   assumes cong: "\<And>f g. simple_function M f \<Longrightarrow> simple_function M g \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> P f \<Longrightarrow> P g"
393   assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
394   assumes mult: "\<And>u c. 0 \<le> c \<Longrightarrow> simple_function M u \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> P (\<lambda>x. c * u x)"
395   assumes add: "\<And>u v. simple_function M u \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> simple_function M v \<Longrightarrow> (\<And>x. 0 \<le> v x) \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
396   shows "P u"
397 proof -
398   show ?thesis
399   proof (rule cong)
400     fix x assume x: "x \<in> space M"
401     from simple_function_indicator_representation[OF u x]
402     show "(\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x) = u x" ..
403   next
404     show "simple_function M (\<lambda>x. (\<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x))"
405       apply (subst simple_function_cong)
406       apply (rule simple_function_indicator_representation[symmetric])
407       apply (auto intro: u)
408       done
409   next
410     from u nn have "finite (u ` space M)" "\<And>x. x \<in> u ` space M \<Longrightarrow> 0 \<le> x"
411       unfolding simple_function_def by auto
412     then show "P (\<lambda>x. \<Sum>y\<in>u ` space M. y * indicator (u -` {y} \<inter> space M) x)"
413     proof induct
414       case empty show ?case
415         using set[of "{}"] by (simp add: indicator_def[abs_def])
416     qed (auto intro!: add mult set simple_functionD u setsum_nonneg
417        simple_function_setsum)
418   qed fact
419 qed
421 lemma borel_measurable_induct[consumes 2, case_names cong set mult add seq, induct set: borel_measurable]:
422   fixes u :: "'a \<Rightarrow> ereal"
423   assumes u: "u \<in> borel_measurable M" "\<And>x. 0 \<le> u x"
424   assumes cong: "\<And>f g. f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> (\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> P g \<Longrightarrow> P f"
425   assumes set: "\<And>A. A \<in> sets M \<Longrightarrow> P (indicator A)"
426   assumes mult: "\<And>u c. 0 \<le> c \<Longrightarrow> u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> P (\<lambda>x. c * u x)"
427   assumes add: "\<And>u v. u \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> u x) \<Longrightarrow> P u \<Longrightarrow> v \<in> borel_measurable M \<Longrightarrow> (\<And>x. 0 \<le> v x) \<Longrightarrow> P v \<Longrightarrow> P (\<lambda>x. v x + u x)"
428   assumes seq: "\<And>U. (\<And>i. U i \<in> borel_measurable M) \<Longrightarrow>  (\<And>i x. 0 \<le> U i x) \<Longrightarrow>  (\<And>i. P (U i)) \<Longrightarrow> incseq U \<Longrightarrow> P (SUP i. U i)"
429   shows "P u"
430   using u
431 proof (induct rule: borel_measurable_implies_simple_function_sequence')
432   fix U assume U: "\<And>i. simple_function M (U i)" "incseq U" "\<And>i. \<infinity> \<notin> range (U i)" and
433     sup: "\<And>x. (SUP i. U i x) = max 0 (u x)" and nn: "\<And>i x. 0 \<le> U i x"
434   have u_eq: "u = (SUP i. U i)"
435     using nn u sup by (auto simp: max_def)
437   from U have "\<And>i. U i \<in> borel_measurable M"
438     by (simp add: borel_measurable_simple_function)
440   show "P u"
441     unfolding u_eq
442   proof (rule seq)
443     fix i show "P (U i)"
444       using `simple_function M (U i)` nn
445       by (induct rule: simple_function_induct_nn)
446          (auto intro: set mult add cong dest!: borel_measurable_simple_function)
447   qed fact+
448 qed
450 lemma simple_function_If_set:
451   assumes sf: "simple_function M f" "simple_function M g" and A: "A \<inter> space M \<in> sets M"
452   shows "simple_function M (\<lambda>x. if x \<in> A then f x else g x)" (is "simple_function M ?IF")
453 proof -
454   def F \<equiv> "\<lambda>x. f -` {x} \<inter> space M" and G \<equiv> "\<lambda>x. g -` {x} \<inter> space M"
455   show ?thesis unfolding simple_function_def
456   proof safe
457     have "?IF ` space M \<subseteq> f ` space M \<union> g ` space M" by auto
458     from finite_subset[OF this] assms
459     show "finite (?IF ` space M)" unfolding simple_function_def by auto
460   next
461     fix x assume "x \<in> space M"
462     then have *: "?IF -` {?IF x} \<inter> space M = (if x \<in> A
463       then ((F (f x) \<inter> (A \<inter> space M)) \<union> (G (f x) - (G (f x) \<inter> (A \<inter> space M))))
464       else ((F (g x) \<inter> (A \<inter> space M)) \<union> (G (g x) - (G (g x) \<inter> (A \<inter> space M)))))"
465       using sets_into_space[OF A] by (auto split: split_if_asm simp: G_def F_def)
466     have [intro]: "\<And>x. F x \<in> sets M" "\<And>x. G x \<in> sets M"
467       unfolding F_def G_def using sf[THEN simple_functionD(2)] by auto
468     show "?IF -` {?IF x} \<inter> space M \<in> sets M" unfolding * using A by auto
469   qed
470 qed
472 lemma simple_function_If:
473   assumes sf: "simple_function M f" "simple_function M g" and P: "{x\<in>space M. P x} \<in> sets M"
474   shows "simple_function M (\<lambda>x. if P x then f x else g x)"
475 proof -
476   have "{x\<in>space M. P x} = {x. P x} \<inter> space M" by auto
477   with simple_function_If_set[OF sf, of "{x. P x}"] P show ?thesis by simp
478 qed
480 lemma simple_function_subalgebra:
481   assumes "simple_function N f"
482   and N_subalgebra: "sets N \<subseteq> sets M" "space N = space M"
483   shows "simple_function M f"
484   using assms unfolding simple_function_def by auto
486 lemma simple_function_comp:
487   assumes T: "T \<in> measurable M M'"
488     and f: "simple_function M' f"
489   shows "simple_function M (\<lambda>x. f (T x))"
490 proof (intro simple_function_def[THEN iffD2] conjI ballI)
491   have "(\<lambda>x. f (T x)) ` space M \<subseteq> f ` space M'"
492     using T unfolding measurable_def by auto
493   then show "finite ((\<lambda>x. f (T x)) ` space M)"
494     using f unfolding simple_function_def by (auto intro: finite_subset)
495   fix i assume i: "i \<in> (\<lambda>x. f (T x)) ` space M"
496   then have "i \<in> f ` space M'"
497     using T unfolding measurable_def by auto
498   then have "f -` {i} \<inter> space M' \<in> sets M'"
499     using f unfolding simple_function_def by auto
500   then have "T -` (f -` {i} \<inter> space M') \<inter> space M \<in> sets M"
501     using T unfolding measurable_def by auto
502   also have "T -` (f -` {i} \<inter> space M') \<inter> space M = (\<lambda>x. f (T x)) -` {i} \<inter> space M"
503     using T unfolding measurable_def by auto
504   finally show "(\<lambda>x. f (T x)) -` {i} \<inter> space M \<in> sets M" .
505 qed
507 section "Simple integral"
509 definition simple_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^isup>S") where
510   "integral\<^isup>S M f = (\<Sum>x \<in> f ` space M. x * emeasure M (f -` {x} \<inter> space M))"
512 syntax
513   "_simple_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<integral>\<^isup>S _. _ \<partial>_" [60,61] 110)
515 translations
516   "\<integral>\<^isup>S x. f \<partial>M" == "CONST simple_integral M (%x. f)"
518 lemma simple_integral_cong:
519   assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
520   shows "integral\<^isup>S M f = integral\<^isup>S M g"
521 proof -
522   have "f ` space M = g ` space M"
523     "\<And>x. f -` {x} \<inter> space M = g -` {x} \<inter> space M"
524     using assms by (auto intro!: image_eqI)
525   thus ?thesis unfolding simple_integral_def by simp
526 qed
528 lemma simple_integral_const[simp]:
529   "(\<integral>\<^isup>Sx. c \<partial>M) = c * (emeasure M) (space M)"
530 proof (cases "space M = {}")
531   case True thus ?thesis unfolding simple_integral_def by simp
532 next
533   case False hence "(\<lambda>x. c) ` space M = {c}" by auto
534   thus ?thesis unfolding simple_integral_def by simp
535 qed
537 lemma simple_function_partition:
538   assumes f: "simple_function M f" and g: "simple_function M g"
539   shows "integral\<^isup>S M f = (\<Sum>A\<in>(\<lambda>x. f -` {f x} \<inter> g -` {g x} \<inter> space M) ` space M. the_elem (f`A) * (emeasure M) A)"
540     (is "_ = setsum _ (?p ` space M)")
541 proof-
542   let ?sub = "\<lambda>x. ?p ` (f -` {x} \<inter> space M)"
543   let ?SIGMA = "Sigma (f`space M) ?sub"
545   have [intro]:
546     "finite (f ` space M)"
547     "finite (g ` space M)"
548     using assms unfolding simple_function_def by simp_all
550   { fix A
551     have "?p ` (A \<inter> space M) \<subseteq>
552       (\<lambda>(x,y). f -` {x} \<inter> g -` {y} \<inter> space M) ` (f`space M \<times> g`space M)"
553       by auto
554     hence "finite (?p ` (A \<inter> space M))"
555       by (rule finite_subset) auto }
556   note this[intro, simp]
557   note sets = simple_function_measurable2[OF f g]
559   { fix x assume "x \<in> space M"
560     have "\<Union>(?sub (f x)) = (f -` {f x} \<inter> space M)" by auto
561     with sets have "(emeasure M) (f -` {f x} \<inter> space M) = setsum (emeasure M) (?sub (f x))"
562       by (subst setsum_emeasure) (auto simp: disjoint_family_on_def) }
563   hence "integral\<^isup>S M f = (\<Sum>(x,A)\<in>?SIGMA. x * (emeasure M) A)"
564     unfolding simple_integral_def using f sets
565     by (subst setsum_Sigma[symmetric])
566        (auto intro!: setsum_cong setsum_ereal_right_distrib)
567   also have "\<dots> = (\<Sum>A\<in>?p ` space M. the_elem (f`A) * (emeasure M) A)"
568   proof -
569     have [simp]: "\<And>x. x \<in> space M \<Longrightarrow> f ` ?p x = {f x}" by (auto intro!: imageI)
570     have "(\<lambda>A. (the_elem (f ` A), A)) ` ?p ` space M
571       = (\<lambda>x. (f x, ?p x)) ` space M"
572     proof safe
573       fix x assume "x \<in> space M"
574       thus "(f x, ?p x) \<in> (\<lambda>A. (the_elem (f`A), A)) ` ?p ` space M"
575         by (auto intro!: image_eqI[of _ _ "?p x"])
576     qed auto
577     thus ?thesis
578       apply (auto intro!: setsum_reindex_cong[of "\<lambda>A. (the_elem (f`A), A)"] inj_onI)
579       apply (rule_tac x="xa" in image_eqI)
580       by simp_all
581   qed
582   finally show ?thesis .
583 qed
586   assumes f: "simple_function M f" and "\<And>x. 0 \<le> f x" and g: "simple_function M g" and "\<And>x. 0 \<le> g x"
587   shows "(\<integral>\<^isup>Sx. f x + g x \<partial>M) = integral\<^isup>S M f + integral\<^isup>S M g"
588 proof -
589   { fix x let ?S = "g -` {g x} \<inter> f -` {f x} \<inter> space M"
590     assume "x \<in> space M"
591     hence "(\<lambda>a. f a + g a) ` ?S = {f x + g x}" "f ` ?S = {f x}" "g ` ?S = {g x}"
592         "(\<lambda>x. (f x, g x)) -` {(f x, g x)} \<inter> space M = ?S"
593       by auto }
594   with assms show ?thesis
595     unfolding
596       simple_function_partition[OF simple_function_add[OF f g] simple_function_Pair[OF f g]]
597       simple_function_partition[OF f g]
598       simple_function_partition[OF g f]
599     by (subst (3) Int_commute)
600        (auto simp add: ereal_left_distrib setsum_addf[symmetric] intro!: setsum_cong)
601 qed
603 lemma simple_integral_setsum[simp]:
604   assumes "\<And>i x. i \<in> P \<Longrightarrow> 0 \<le> f i x"
605   assumes "\<And>i. i \<in> P \<Longrightarrow> simple_function M (f i)"
606   shows "(\<integral>\<^isup>Sx. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^isup>S M (f i))"
607 proof cases
608   assume "finite P"
609   from this assms show ?thesis
610     by induct (auto simp: simple_function_setsum simple_integral_add setsum_nonneg)
611 qed auto
613 lemma simple_integral_mult[simp]:
614   assumes f: "simple_function M f" "\<And>x. 0 \<le> f x"
615   shows "(\<integral>\<^isup>Sx. c * f x \<partial>M) = c * integral\<^isup>S M f"
616 proof -
617   note mult = simple_function_mult[OF simple_function_const[of _ c] f(1)]
618   { fix x let ?S = "f -` {f x} \<inter> (\<lambda>x. c * f x) -` {c * f x} \<inter> space M"
619     assume "x \<in> space M"
620     hence "(\<lambda>x. c * f x) ` ?S = {c * f x}" "f ` ?S = {f x}"
621       by auto }
622   with assms show ?thesis
623     unfolding simple_function_partition[OF mult f(1)]
624               simple_function_partition[OF f(1) mult]
625     by (subst setsum_ereal_right_distrib)
626        (auto intro!: ereal_0_le_mult setsum_cong simp: mult_assoc)
627 qed
629 lemma simple_integral_mono_AE:
630   assumes f: "simple_function M f" and g: "simple_function M g"
631   and mono: "AE x in M. f x \<le> g x"
632   shows "integral\<^isup>S M f \<le> integral\<^isup>S M g"
633 proof -
634   let ?S = "\<lambda>x. (g -` {g x} \<inter> space M) \<inter> (f -` {f x} \<inter> space M)"
635   have *: "\<And>x. g -` {g x} \<inter> f -` {f x} \<inter> space M = ?S x"
636     "\<And>x. f -` {f x} \<inter> g -` {g x} \<inter> space M = ?S x" by auto
637   show ?thesis
638     unfolding *
639       simple_function_partition[OF f g]
640       simple_function_partition[OF g f]
641   proof (safe intro!: setsum_mono)
642     fix x assume "x \<in> space M"
643     then have *: "f ` ?S x = {f x}" "g ` ?S x = {g x}" by auto
644     show "the_elem (f`?S x) * (emeasure M) (?S x) \<le> the_elem (g`?S x) * (emeasure M) (?S x)"
645     proof (cases "f x \<le> g x")
646       case True then show ?thesis
647         using * assms(1,2)[THEN simple_functionD(2)]
648         by (auto intro!: ereal_mult_right_mono)
649     next
650       case False
651       obtain N where N: "{x\<in>space M. \<not> f x \<le> g x} \<subseteq> N" "N \<in> sets M" "(emeasure M) N = 0"
652         using mono by (auto elim!: AE_E)
653       have "?S x \<subseteq> N" using N `x \<in> space M` False by auto
654       moreover have "?S x \<in> sets M" using assms
655         by (rule_tac Int) (auto intro!: simple_functionD)
656       ultimately have "(emeasure M) (?S x) \<le> (emeasure M) N"
657         using `N \<in> sets M` by (auto intro!: emeasure_mono)
658       moreover have "0 \<le> (emeasure M) (?S x)"
659         using assms(1,2)[THEN simple_functionD(2)] by auto
660       ultimately have "(emeasure M) (?S x) = 0" using `(emeasure M) N = 0` by auto
661       then show ?thesis by simp
662     qed
663   qed
664 qed
666 lemma simple_integral_mono:
667   assumes "simple_function M f" and "simple_function M g"
668   and mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
669   shows "integral\<^isup>S M f \<le> integral\<^isup>S M g"
670   using assms by (intro simple_integral_mono_AE) auto
672 lemma simple_integral_cong_AE:
673   assumes "simple_function M f" and "simple_function M g"
674   and "AE x in M. f x = g x"
675   shows "integral\<^isup>S M f = integral\<^isup>S M g"
676   using assms by (auto simp: eq_iff intro!: simple_integral_mono_AE)
678 lemma simple_integral_cong':
679   assumes sf: "simple_function M f" "simple_function M g"
680   and mea: "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0"
681   shows "integral\<^isup>S M f = integral\<^isup>S M g"
682 proof (intro simple_integral_cong_AE sf AE_I)
683   show "(emeasure M) {x\<in>space M. f x \<noteq> g x} = 0" by fact
684   show "{x \<in> space M. f x \<noteq> g x} \<in> sets M"
685     using sf[THEN borel_measurable_simple_function] by auto
686 qed simp
688 lemma simple_integral_indicator:
689   assumes "A \<in> sets M"
690   assumes f: "simple_function M f"
691   shows "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) =
692     (\<Sum>x \<in> f ` space M. x * (emeasure M) (f -` {x} \<inter> space M \<inter> A))"
693 proof cases
694   assume "A = space M"
695   moreover hence "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) = integral\<^isup>S M f"
696     by (auto intro!: simple_integral_cong)
697   moreover have "\<And>X. X \<inter> space M \<inter> space M = X \<inter> space M" by auto
698   ultimately show ?thesis by (simp add: simple_integral_def)
699 next
700   assume "A \<noteq> space M"
701   then obtain x where x: "x \<in> space M" "x \<notin> A" using sets_into_space[OF assms(1)] by auto
702   have I: "(\<lambda>x. f x * indicator A x) ` space M = f ` A \<union> {0}" (is "?I ` _ = _")
703   proof safe
704     fix y assume "?I y \<notin> f ` A" hence "y \<notin> A" by auto thus "?I y = 0" by auto
705   next
706     fix y assume "y \<in> A" thus "f y \<in> ?I ` space M"
707       using sets_into_space[OF assms(1)] by (auto intro!: image_eqI[of _ _ y])
708   next
709     show "0 \<in> ?I ` space M" using x by (auto intro!: image_eqI[of _ _ x])
710   qed
711   have *: "(\<integral>\<^isup>Sx. f x * indicator A x \<partial>M) =
712     (\<Sum>x \<in> f ` space M \<union> {0}. x * (emeasure M) (f -` {x} \<inter> space M \<inter> A))"
713     unfolding simple_integral_def I
714   proof (rule setsum_mono_zero_cong_left)
715     show "finite (f ` space M \<union> {0})"
716       using assms(2) unfolding simple_function_def by auto
717     show "f ` A \<union> {0} \<subseteq> f`space M \<union> {0}"
718       using sets_into_space[OF assms(1)] by auto
719     have "\<And>x. f x \<notin> f ` A \<Longrightarrow> f -` {f x} \<inter> space M \<inter> A = {}"
720       by (auto simp: image_iff)
721     thus "\<forall>i\<in>f ` space M \<union> {0} - (f ` A \<union> {0}).
722       i * (emeasure M) (f -` {i} \<inter> space M \<inter> A) = 0" by auto
723   next
724     fix x assume "x \<in> f`A \<union> {0}"
725     hence "x \<noteq> 0 \<Longrightarrow> ?I -` {x} \<inter> space M = f -` {x} \<inter> space M \<inter> A"
726       by (auto simp: indicator_def split: split_if_asm)
727     thus "x * (emeasure M) (?I -` {x} \<inter> space M) =
728       x * (emeasure M) (f -` {x} \<inter> space M \<inter> A)" by (cases "x = 0") simp_all
729   qed
730   show ?thesis unfolding *
731     using assms(2) unfolding simple_function_def
732     by (auto intro!: setsum_mono_zero_cong_right)
733 qed
735 lemma simple_integral_indicator_only[simp]:
736   assumes "A \<in> sets M"
737   shows "integral\<^isup>S M (indicator A) = emeasure M A"
738 proof cases
739   assume "space M = {}" hence "A = {}" using sets_into_space[OF assms] by auto
740   thus ?thesis unfolding simple_integral_def using `space M = {}` by auto
741 next
742   assume "space M \<noteq> {}" hence "(\<lambda>x. 1) ` space M = {1::ereal}" by auto
743   thus ?thesis
744     using simple_integral_indicator[OF assms simple_function_const[of _ 1]]
745     using sets_into_space[OF assms]
746     by (auto intro!: arg_cong[where f="(emeasure M)"])
747 qed
749 lemma simple_integral_null_set:
750   assumes "simple_function M u" "\<And>x. 0 \<le> u x" and "N \<in> null_sets M"
751   shows "(\<integral>\<^isup>Sx. u x * indicator N x \<partial>M) = 0"
752 proof -
753   have "AE x in M. indicator N x = (0 :: ereal)"
754     using `N \<in> null_sets M` by (auto simp: indicator_def intro!: AE_I[of _ _ N])
755   then have "(\<integral>\<^isup>Sx. u x * indicator N x \<partial>M) = (\<integral>\<^isup>Sx. 0 \<partial>M)"
756     using assms apply (intro simple_integral_cong_AE) by auto
757   then show ?thesis by simp
758 qed
760 lemma simple_integral_cong_AE_mult_indicator:
761   assumes sf: "simple_function M f" and eq: "AE x in M. x \<in> S" and "S \<in> sets M"
762   shows "integral\<^isup>S M f = (\<integral>\<^isup>Sx. f x * indicator S x \<partial>M)"
763   using assms by (intro simple_integral_cong_AE) auto
765 lemma simple_integral_cmult_indicator:
766   assumes A: "A \<in> sets M"
767   shows "(\<integral>\<^isup>Sx. c * indicator A x \<partial>M) = c * (emeasure M) A"
768   using simple_integral_mult[OF simple_function_indicator[OF A]]
769   unfolding simple_integral_indicator_only[OF A] by simp
771 lemma simple_integral_positive:
772   assumes f: "simple_function M f" and ae: "AE x in M. 0 \<le> f x"
773   shows "0 \<le> integral\<^isup>S M f"
774 proof -
775   have "integral\<^isup>S M (\<lambda>x. 0) \<le> integral\<^isup>S M f"
776     using simple_integral_mono_AE[OF _ f ae] by auto
777   then show ?thesis by simp
778 qed
780 section "Continuous positive integration"
782 definition positive_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> ereal" ("integral\<^isup>P") where
783   "integral\<^isup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f}. integral\<^isup>S M g)"
785 syntax
786   "_positive_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> 'a measure \<Rightarrow> ereal" ("\<integral>\<^isup>+ _. _ \<partial>_" [60,61] 110)
788 translations
789   "\<integral>\<^isup>+ x. f \<partial>M" == "CONST positive_integral M (%x. f)"
791 lemma positive_integral_positive:
792   "0 \<le> integral\<^isup>P M f"
793   by (auto intro!: SUP_upper2[of "\<lambda>x. 0"] simp: positive_integral_def le_fun_def)
795 lemma positive_integral_not_MInfty[simp]: "integral\<^isup>P M f \<noteq> -\<infinity>"
796   using positive_integral_positive[of M f] by auto
798 lemma positive_integral_def_finite:
799   "integral\<^isup>P M f = (SUP g : {g. simple_function M g \<and> g \<le> max 0 \<circ> f \<and> range g \<subseteq> {0 ..< \<infinity>}}. integral\<^isup>S M g)"
800     (is "_ = SUPR ?A ?f")
801   unfolding positive_integral_def
802 proof (safe intro!: antisym SUP_least)
803   fix g assume g: "simple_function M g" "g \<le> max 0 \<circ> f"
804   let ?G = "{x \<in> space M. \<not> g x \<noteq> \<infinity>}"
805   note gM = g(1)[THEN borel_measurable_simple_function]
806   have \<mu>G_pos: "0 \<le> (emeasure M) ?G" using gM by auto
807   let ?g = "\<lambda>y x. if g x = \<infinity> then y else max 0 (g x)"
808   from g gM have g_in_A: "\<And>y. 0 \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> ?g y \<in> ?A"
809     apply (safe intro!: simple_function_max simple_function_If)
810     apply (force simp: max_def le_fun_def split: split_if_asm)+
811     done
812   show "integral\<^isup>S M g \<le> SUPR ?A ?f"
813   proof cases
814     have g0: "?g 0 \<in> ?A" by (intro g_in_A) auto
815     assume "(emeasure M) ?G = 0"
816     with gM have "AE x in M. x \<notin> ?G"
817       by (auto simp add: AE_iff_null intro!: null_setsI)
818     with gM g show ?thesis
819       by (intro SUP_upper2[OF g0] simple_integral_mono_AE)
820          (auto simp: max_def intro!: simple_function_If)
821   next
822     assume \<mu>G: "(emeasure M) ?G \<noteq> 0"
823     have "SUPR ?A (integral\<^isup>S M) = \<infinity>"
824     proof (intro SUP_PInfty)
825       fix n :: nat
826       let ?y = "ereal (real n) / (if (emeasure M) ?G = \<infinity> then 1 else (emeasure M) ?G)"
827       have "0 \<le> ?y" "?y \<noteq> \<infinity>" using \<mu>G \<mu>G_pos by (auto simp: ereal_divide_eq)
828       then have "?g ?y \<in> ?A" by (rule g_in_A)
829       have "real n \<le> ?y * (emeasure M) ?G"
830         using \<mu>G \<mu>G_pos by (cases "(emeasure M) ?G") (auto simp: field_simps)
831       also have "\<dots> = (\<integral>\<^isup>Sx. ?y * indicator ?G x \<partial>M)"
832         using `0 \<le> ?y` `?g ?y \<in> ?A` gM
833         by (subst simple_integral_cmult_indicator) auto
834       also have "\<dots> \<le> integral\<^isup>S M (?g ?y)" using `?g ?y \<in> ?A` gM
835         by (intro simple_integral_mono) auto
836       finally show "\<exists>i\<in>?A. real n \<le> integral\<^isup>S M i"
837         using `?g ?y \<in> ?A` by blast
838     qed
839     then show ?thesis by simp
840   qed
841 qed (auto intro: SUP_upper)
843 lemma positive_integral_mono_AE:
844   assumes ae: "AE x in M. u x \<le> v x" shows "integral\<^isup>P M u \<le> integral\<^isup>P M v"
845   unfolding positive_integral_def
846 proof (safe intro!: SUP_mono)
847   fix n assume n: "simple_function M n" "n \<le> max 0 \<circ> u"
848   from ae[THEN AE_E] guess N . note N = this
849   then have ae_N: "AE x in M. x \<notin> N" by (auto intro: AE_not_in)
850   let ?n = "\<lambda>x. n x * indicator (space M - N) x"
851   have "AE x in M. n x \<le> ?n x" "simple_function M ?n"
852     using n N ae_N by auto
853   moreover
854   { fix x have "?n x \<le> max 0 (v x)"
855     proof cases
856       assume x: "x \<in> space M - N"
857       with N have "u x \<le> v x" by auto
858       with n(2)[THEN le_funD, of x] x show ?thesis
859         by (auto simp: max_def split: split_if_asm)
860     qed simp }
861   then have "?n \<le> max 0 \<circ> v" by (auto simp: le_funI)
862   moreover have "integral\<^isup>S M n \<le> integral\<^isup>S M ?n"
863     using ae_N N n by (auto intro!: simple_integral_mono_AE)
864   ultimately show "\<exists>m\<in>{g. simple_function M g \<and> g \<le> max 0 \<circ> v}. integral\<^isup>S M n \<le> integral\<^isup>S M m"
865     by force
866 qed
868 lemma positive_integral_mono:
869   "(\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x) \<Longrightarrow> integral\<^isup>P M u \<le> integral\<^isup>P M v"
870   by (auto intro: positive_integral_mono_AE)
872 lemma positive_integral_cong_AE:
873   "AE x in M. u x = v x \<Longrightarrow> integral\<^isup>P M u = integral\<^isup>P M v"
874   by (auto simp: eq_iff intro!: positive_integral_mono_AE)
876 lemma positive_integral_cong:
877   "(\<And>x. x \<in> space M \<Longrightarrow> u x = v x) \<Longrightarrow> integral\<^isup>P M u = integral\<^isup>P M v"
878   by (auto intro: positive_integral_cong_AE)
880 lemma positive_integral_eq_simple_integral:
881   assumes f: "simple_function M f" "\<And>x. 0 \<le> f x" shows "integral\<^isup>P M f = integral\<^isup>S M f"
882 proof -
883   let ?f = "\<lambda>x. f x * indicator (space M) x"
884   have f': "simple_function M ?f" using f by auto
885   with f(2) have [simp]: "max 0 \<circ> ?f = ?f"
886     by (auto simp: fun_eq_iff max_def split: split_indicator)
887   have "integral\<^isup>P M ?f \<le> integral\<^isup>S M ?f" using f'
888     by (force intro!: SUP_least simple_integral_mono simp: le_fun_def positive_integral_def)
889   moreover have "integral\<^isup>S M ?f \<le> integral\<^isup>P M ?f"
890     unfolding positive_integral_def
891     using f' by (auto intro!: SUP_upper)
892   ultimately show ?thesis
893     by (simp cong: positive_integral_cong simple_integral_cong)
894 qed
896 lemma positive_integral_eq_simple_integral_AE:
897   assumes f: "simple_function M f" "AE x in M. 0 \<le> f x" shows "integral\<^isup>P M f = integral\<^isup>S M f"
898 proof -
899   have "AE x in M. f x = max 0 (f x)" using f by (auto split: split_max)
900   with f have "integral\<^isup>P M f = integral\<^isup>S M (\<lambda>x. max 0 (f x))"
901     by (simp cong: positive_integral_cong_AE simple_integral_cong_AE
903   with assms show ?thesis
904     by (auto intro!: simple_integral_cong_AE split: split_max)
905 qed
907 lemma positive_integral_SUP_approx:
908   assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
909   and u: "simple_function M u" "u \<le> (SUP i. f i)" "u`space M \<subseteq> {0..<\<infinity>}"
910   shows "integral\<^isup>S M u \<le> (SUP i. integral\<^isup>P M (f i))" (is "_ \<le> ?S")
911 proof (rule ereal_le_mult_one_interval)
912   have "0 \<le> (SUP i. integral\<^isup>P M (f i))"
913     using f(3) by (auto intro!: SUP_upper2 positive_integral_positive)
914   then show "(SUP i. integral\<^isup>P M (f i)) \<noteq> -\<infinity>" by auto
915   have u_range: "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> u x \<and> u x \<noteq> \<infinity>"
916     using u(3) by auto
917   fix a :: ereal assume "0 < a" "a < 1"
918   hence "a \<noteq> 0" by auto
919   let ?B = "\<lambda>i. {x \<in> space M. a * u x \<le> f i x}"
920   have B: "\<And>i. ?B i \<in> sets M"
921     using f `simple_function M u` by (auto simp: borel_measurable_simple_function)
923   let ?uB = "\<lambda>i x. u x * indicator (?B i) x"
925   { fix i have "?B i \<subseteq> ?B (Suc i)"
926     proof safe
927       fix i x assume "a * u x \<le> f i x"
928       also have "\<dots> \<le> f (Suc i) x"
929         using `incseq f`[THEN incseq_SucD] unfolding le_fun_def by auto
930       finally show "a * u x \<le> f (Suc i) x" .
931     qed }
932   note B_mono = this
934   note B_u = Int[OF u(1)[THEN simple_functionD(2)] B]
936   let ?B' = "\<lambda>i n. (u -` {i} \<inter> space M) \<inter> ?B n"
937   have measure_conv: "\<And>i. (emeasure M) (u -` {i} \<inter> space M) = (SUP n. (emeasure M) (?B' i n))"
938   proof -
939     fix i
940     have 1: "range (?B' i) \<subseteq> sets M" using B_u by auto
941     have 2: "incseq (?B' i)" using B_mono by (auto intro!: incseq_SucI)
942     have "(\<Union>n. ?B' i n) = u -` {i} \<inter> space M"
943     proof safe
944       fix x i assume x: "x \<in> space M"
945       show "x \<in> (\<Union>i. ?B' (u x) i)"
946       proof cases
947         assume "u x = 0" thus ?thesis using `x \<in> space M` f(3) by simp
948       next
949         assume "u x \<noteq> 0"
950         with `a < 1` u_range[OF `x \<in> space M`]
951         have "a * u x < 1 * u x"
952           by (intro ereal_mult_strict_right_mono) (auto simp: image_iff)
953         also have "\<dots> \<le> (SUP i. f i x)" using u(2) by (auto simp: le_fun_def)
954         finally obtain i where "a * u x < f i x" unfolding SUP_def
955           by (auto simp add: less_Sup_iff)
956         hence "a * u x \<le> f i x" by auto
957         thus ?thesis using `x \<in> space M` by auto
958       qed
959     qed
960     then show "?thesis i" using SUP_emeasure_incseq[OF 1 2] by simp
961   qed
963   have "integral\<^isup>S M u = (SUP i. integral\<^isup>S M (?uB i))"
964     unfolding simple_integral_indicator[OF B `simple_function M u`]
965   proof (subst SUPR_ereal_setsum, safe)
966     fix x n assume "x \<in> space M"
967     with u_range show "incseq (\<lambda>i. u x * (emeasure M) (?B' (u x) i))" "\<And>i. 0 \<le> u x * (emeasure M) (?B' (u x) i)"
968       using B_mono B_u by (auto intro!: emeasure_mono ereal_mult_left_mono incseq_SucI simp: ereal_zero_le_0_iff)
969   next
970     show "integral\<^isup>S M u = (\<Sum>i\<in>u ` space M. SUP n. i * (emeasure M) (?B' i n))"
971       using measure_conv u_range B_u unfolding simple_integral_def
972       by (auto intro!: setsum_cong SUPR_ereal_cmult[symmetric])
973   qed
974   moreover
975   have "a * (SUP i. integral\<^isup>S M (?uB i)) \<le> ?S"
976     apply (subst SUPR_ereal_cmult[symmetric])
977   proof (safe intro!: SUP_mono bexI)
978     fix i
979     have "a * integral\<^isup>S M (?uB i) = (\<integral>\<^isup>Sx. a * ?uB i x \<partial>M)"
980       using B `simple_function M u` u_range
981       by (subst simple_integral_mult) (auto split: split_indicator)
982     also have "\<dots> \<le> integral\<^isup>P M (f i)"
983     proof -
984       have *: "simple_function M (\<lambda>x. a * ?uB i x)" using B `0 < a` u(1) by auto
985       show ?thesis using f(3) * u_range `0 < a`
986         by (subst positive_integral_eq_simple_integral[symmetric])
987            (auto intro!: positive_integral_mono split: split_indicator)
988     qed
989     finally show "a * integral\<^isup>S M (?uB i) \<le> integral\<^isup>P M (f i)"
990       by auto
991   next
992     fix i show "0 \<le> \<integral>\<^isup>S x. ?uB i x \<partial>M" using B `0 < a` u(1) u_range
993       by (intro simple_integral_positive) (auto split: split_indicator)
994   qed (insert `0 < a`, auto)
995   ultimately show "a * integral\<^isup>S M u \<le> ?S" by simp
996 qed
998 lemma incseq_positive_integral:
999   assumes "incseq f" shows "incseq (\<lambda>i. integral\<^isup>P M (f i))"
1000 proof -
1001   have "\<And>i x. f i x \<le> f (Suc i) x"
1002     using assms by (auto dest!: incseq_SucD simp: le_fun_def)
1003   then show ?thesis
1004     by (auto intro!: incseq_SucI positive_integral_mono)
1005 qed
1007 text {* Beppo-Levi monotone convergence theorem *}
1008 lemma positive_integral_monotone_convergence_SUP:
1009   assumes f: "incseq f" "\<And>i. f i \<in> borel_measurable M" "\<And>i x. 0 \<le> f i x"
1010   shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
1011 proof (rule antisym)
1012   show "(SUP j. integral\<^isup>P M (f j)) \<le> (\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M)"
1013     by (auto intro!: SUP_least SUP_upper positive_integral_mono)
1014 next
1015   show "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) \<le> (SUP j. integral\<^isup>P M (f j))"
1016     unfolding positive_integral_def_finite[of _ "\<lambda>x. SUP i. f i x"]
1017   proof (safe intro!: SUP_least)
1018     fix g assume g: "simple_function M g"
1019       and "g \<le> max 0 \<circ> (\<lambda>x. SUP i. f i x)" "range g \<subseteq> {0..<\<infinity>}"
1020     moreover then have "\<And>x. 0 \<le> (SUP i. f i x)" and g': "g`space M \<subseteq> {0..<\<infinity>}"
1021       using f by (auto intro!: SUP_upper2)
1022     ultimately show "integral\<^isup>S M g \<le> (SUP j. integral\<^isup>P M (f j))"
1023       by (intro  positive_integral_SUP_approx[OF f g _ g'])
1024          (auto simp: le_fun_def max_def)
1025   qed
1026 qed
1028 lemma positive_integral_monotone_convergence_SUP_AE:
1029   assumes f: "\<And>i. AE x in M. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x" "\<And>i. f i \<in> borel_measurable M"
1030   shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
1031 proof -
1032   from f have "AE x in M. \<forall>i. f i x \<le> f (Suc i) x \<and> 0 \<le> f i x"
1033     by (simp add: AE_all_countable)
1034   from this[THEN AE_E] guess N . note N = this
1035   let ?f = "\<lambda>i x. if x \<in> space M - N then f i x else 0"
1036   have f_eq: "AE x in M. \<forall>i. ?f i x = f i x" using N by (auto intro!: AE_I[of _ _ N])
1037   then have "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (\<integral>\<^isup>+ x. (SUP i. ?f i x) \<partial>M)"
1038     by (auto intro!: positive_integral_cong_AE)
1039   also have "\<dots> = (SUP i. (\<integral>\<^isup>+ x. ?f i x \<partial>M))"
1040   proof (rule positive_integral_monotone_convergence_SUP)
1041     show "incseq ?f" using N(1) by (force intro!: incseq_SucI le_funI)
1042     { fix i show "(\<lambda>x. if x \<in> space M - N then f i x else 0) \<in> borel_measurable M"
1043         using f N(3) by (intro measurable_If_set) auto
1044       fix x show "0 \<le> ?f i x"
1045         using N(1) by auto }
1046   qed
1047   also have "\<dots> = (SUP i. (\<integral>\<^isup>+ x. f i x \<partial>M))"
1048     using f_eq by (force intro!: arg_cong[where f="SUPR UNIV"] positive_integral_cong_AE ext)
1049   finally show ?thesis .
1050 qed
1052 lemma positive_integral_monotone_convergence_SUP_AE_incseq:
1053   assumes f: "incseq f" "\<And>i. AE x in M. 0 \<le> f i x" and borel: "\<And>i. f i \<in> borel_measurable M"
1054   shows "(\<integral>\<^isup>+ x. (SUP i. f i x) \<partial>M) = (SUP i. integral\<^isup>P M (f i))"
1055   using f[unfolded incseq_Suc_iff le_fun_def]
1056   by (intro positive_integral_monotone_convergence_SUP_AE[OF _ borel])
1057      auto
1059 lemma positive_integral_monotone_convergence_simple:
1060   assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
1061   shows "(SUP i. integral\<^isup>S M (f i)) = (\<integral>\<^isup>+x. (SUP i. f i x) \<partial>M)"
1062   using assms unfolding positive_integral_monotone_convergence_SUP[OF f(1)
1063     f(3)[THEN borel_measurable_simple_function] f(2)]
1064   by (auto intro!: positive_integral_eq_simple_integral[symmetric] arg_cong[where f="SUPR UNIV"] ext)
1066 lemma positive_integral_max_0:
1067   "(\<integral>\<^isup>+x. max 0 (f x) \<partial>M) = integral\<^isup>P M f"
1068   by (simp add: le_fun_def positive_integral_def)
1070 lemma positive_integral_cong_pos:
1071   assumes "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> 0 \<and> g x \<le> 0 \<or> f x = g x"
1072   shows "integral\<^isup>P M f = integral\<^isup>P M g"
1073 proof -
1074   have "integral\<^isup>P M (\<lambda>x. max 0 (f x)) = integral\<^isup>P M (\<lambda>x. max 0 (g x))"
1075   proof (intro positive_integral_cong)
1076     fix x assume "x \<in> space M"
1077     from assms[OF this] show "max 0 (f x) = max 0 (g x)"
1078       by (auto split: split_max)
1079   qed
1080   then show ?thesis by (simp add: positive_integral_max_0)
1081 qed
1083 lemma SUP_simple_integral_sequences:
1084   assumes f: "incseq f" "\<And>i x. 0 \<le> f i x" "\<And>i. simple_function M (f i)"
1085   and g: "incseq g" "\<And>i x. 0 \<le> g i x" "\<And>i. simple_function M (g i)"
1086   and eq: "AE x in M. (SUP i. f i x) = (SUP i. g i x)"
1087   shows "(SUP i. integral\<^isup>S M (f i)) = (SUP i. integral\<^isup>S M (g i))"
1088     (is "SUPR _ ?F = SUPR _ ?G")
1089 proof -
1090   have "(SUP i. integral\<^isup>S M (f i)) = (\<integral>\<^isup>+x. (SUP i. f i x) \<partial>M)"
1091     using f by (rule positive_integral_monotone_convergence_simple)
1092   also have "\<dots> = (\<integral>\<^isup>+x. (SUP i. g i x) \<partial>M)"
1093     unfolding eq[THEN positive_integral_cong_AE] ..
1094   also have "\<dots> = (SUP i. ?G i)"
1095     using g by (rule positive_integral_monotone_convergence_simple[symmetric])
1096   finally show ?thesis by simp
1097 qed
1099 lemma positive_integral_const[simp]:
1100   "0 \<le> c \<Longrightarrow> (\<integral>\<^isup>+ x. c \<partial>M) = c * (emeasure M) (space M)"
1101   by (subst positive_integral_eq_simple_integral) auto
1103 lemma positive_integral_linear:
1104   assumes f: "f \<in> borel_measurable M" "\<And>x. 0 \<le> f x" and "0 \<le> a"
1105   and g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x"
1106   shows "(\<integral>\<^isup>+ x. a * f x + g x \<partial>M) = a * integral\<^isup>P M f + integral\<^isup>P M g"
1107     (is "integral\<^isup>P M ?L = _")
1108 proof -
1109   from borel_measurable_implies_simple_function_sequence'[OF f(1)] guess u .
1110   note u = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
1111   from borel_measurable_implies_simple_function_sequence'[OF g(1)] guess v .
1112   note v = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
1113   let ?L' = "\<lambda>i x. a * u i x + v i x"
1115   have "?L \<in> borel_measurable M" using assms by auto
1116   from borel_measurable_implies_simple_function_sequence'[OF this] guess l .
1117   note l = positive_integral_monotone_convergence_simple[OF this(2,5,1)] this
1119   have inc: "incseq (\<lambda>i. a * integral\<^isup>S M (u i))" "incseq (\<lambda>i. integral\<^isup>S M (v i))"
1120     using u v `0 \<le> a`
1121     by (auto simp: incseq_Suc_iff le_fun_def
1122              intro!: add_mono ereal_mult_left_mono simple_integral_mono)
1123   have pos: "\<And>i. 0 \<le> integral\<^isup>S M (u i)" "\<And>i. 0 \<le> integral\<^isup>S M (v i)" "\<And>i. 0 \<le> a * integral\<^isup>S M (u i)"
1124     using u v `0 \<le> a` by (auto simp: simple_integral_positive)
1125   { fix i from pos[of i] have "a * integral\<^isup>S M (u i) \<noteq> -\<infinity>" "integral\<^isup>S M (v i) \<noteq> -\<infinity>"
1126       by (auto split: split_if_asm) }
1127   note not_MInf = this
1129   have l': "(SUP i. integral\<^isup>S M (l i)) = (SUP i. integral\<^isup>S M (?L' i))"
1130   proof (rule SUP_simple_integral_sequences[OF l(3,6,2)])
1131     show "incseq ?L'" "\<And>i x. 0 \<le> ?L' i x" "\<And>i. simple_function M (?L' i)"
1132       using u v  `0 \<le> a` unfolding incseq_Suc_iff le_fun_def
1133       by (auto intro!: add_mono ereal_mult_left_mono ereal_add_nonneg_nonneg)
1134     { fix x
1135       { fix i have "a * u i x \<noteq> -\<infinity>" "v i x \<noteq> -\<infinity>" "u i x \<noteq> -\<infinity>" using `0 \<le> a` u(6)[of i x] v(6)[of i x]
1136           by auto }
1137       then have "(SUP i. a * u i x + v i x) = a * (SUP i. u i x) + (SUP i. v i x)"
1138         using `0 \<le> a` u(3) v(3) u(6)[of _ x] v(6)[of _ x]
1139         by (subst SUPR_ereal_cmult[symmetric, OF u(6) `0 \<le> a`])
1140            (auto intro!: SUPR_ereal_add
1141                  simp: incseq_Suc_iff le_fun_def add_mono ereal_mult_left_mono ereal_add_nonneg_nonneg) }
1142     then show "AE x in M. (SUP i. l i x) = (SUP i. ?L' i x)"
1143       unfolding l(5) using `0 \<le> a` u(5) v(5) l(5) f(2) g(2)
1144       by (intro AE_I2) (auto split: split_max simp add: ereal_add_nonneg_nonneg)
1145   qed
1146   also have "\<dots> = (SUP i. a * integral\<^isup>S M (u i) + integral\<^isup>S M (v i))"
1147     using u(2, 6) v(2, 6) `0 \<le> a` by (auto intro!: arg_cong[where f="SUPR UNIV"] ext)
1148   finally have "(\<integral>\<^isup>+ x. max 0 (a * f x + g x) \<partial>M) = a * (\<integral>\<^isup>+x. max 0 (f x) \<partial>M) + (\<integral>\<^isup>+x. max 0 (g x) \<partial>M)"
1149     unfolding l(5)[symmetric] u(5)[symmetric] v(5)[symmetric]
1150     unfolding l(1)[symmetric] u(1)[symmetric] v(1)[symmetric]
1151     apply (subst SUPR_ereal_cmult[symmetric, OF pos(1) `0 \<le> a`])
1152     apply (subst SUPR_ereal_add[symmetric, OF inc not_MInf]) .
1153   then show ?thesis by (simp add: positive_integral_max_0)
1154 qed
1156 lemma positive_integral_cmult:
1157   assumes f: "f \<in> borel_measurable M" "0 \<le> c"
1158   shows "(\<integral>\<^isup>+ x. c * f x \<partial>M) = c * integral\<^isup>P M f"
1159 proof -
1160   have [simp]: "\<And>x. c * max 0 (f x) = max 0 (c * f x)" using `0 \<le> c`
1161     by (auto split: split_max simp: ereal_zero_le_0_iff)
1162   have "(\<integral>\<^isup>+ x. c * f x \<partial>M) = (\<integral>\<^isup>+ x. c * max 0 (f x) \<partial>M)"
1163     by (simp add: positive_integral_max_0)
1164   then show ?thesis
1165     using positive_integral_linear[OF _ _ `0 \<le> c`, of "\<lambda>x. max 0 (f x)" _ "\<lambda>x. 0"] f
1166     by (auto simp: positive_integral_max_0)
1167 qed
1169 lemma positive_integral_multc:
1170   assumes "f \<in> borel_measurable M" "0 \<le> c"
1171   shows "(\<integral>\<^isup>+ x. f x * c \<partial>M) = integral\<^isup>P M f * c"
1172   unfolding mult_commute[of _ c] positive_integral_cmult[OF assms] by simp
1174 lemma positive_integral_indicator[simp]:
1175   "A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. indicator A x\<partial>M) = (emeasure M) A"
1176   by (subst positive_integral_eq_simple_integral)
1177      (auto simp: simple_integral_indicator)
1179 lemma positive_integral_cmult_indicator:
1180   "0 \<le> c \<Longrightarrow> A \<in> sets M \<Longrightarrow> (\<integral>\<^isup>+ x. c * indicator A x \<partial>M) = c * (emeasure M) A"
1181   by (subst positive_integral_eq_simple_integral)
1182      (auto simp: simple_function_indicator simple_integral_indicator)
1185   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
1186   and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
1187   shows "(\<integral>\<^isup>+ x. f x + g x \<partial>M) = integral\<^isup>P M f + integral\<^isup>P M g"
1188 proof -
1189   have ae: "AE x in M. max 0 (f x) + max 0 (g x) = max 0 (f x + g x)"
1190     using assms by (auto split: split_max simp: ereal_add_nonneg_nonneg)
1191   have "(\<integral>\<^isup>+ x. f x + g x \<partial>M) = (\<integral>\<^isup>+ x. max 0 (f x + g x) \<partial>M)"
1192     by (simp add: positive_integral_max_0)
1193   also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) + max 0 (g x) \<partial>M)"
1194     unfolding ae[THEN positive_integral_cong_AE] ..
1195   also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) \<partial>M) + (\<integral>\<^isup>+ x. max 0 (g x) \<partial>M)"
1196     using positive_integral_linear[of "\<lambda>x. max 0 (f x)" _ 1 "\<lambda>x. max 0 (g x)"] f g
1197     by auto
1198   finally show ?thesis
1199     by (simp add: positive_integral_max_0)
1200 qed
1202 lemma positive_integral_setsum:
1203   assumes "\<And>i. i\<in>P \<Longrightarrow> f i \<in> borel_measurable M" "\<And>i. i\<in>P \<Longrightarrow> AE x in M. 0 \<le> f i x"
1204   shows "(\<integral>\<^isup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M) = (\<Sum>i\<in>P. integral\<^isup>P M (f i))"
1205 proof cases
1206   assume f: "finite P"
1207   from assms have "AE x in M. \<forall>i\<in>P. 0 \<le> f i x" unfolding AE_finite_all[OF f] by auto
1208   from f this assms(1) show ?thesis
1209   proof induct
1210     case (insert i P)
1211     then have "f i \<in> borel_measurable M" "AE x in M. 0 \<le> f i x"
1212       "(\<lambda>x. \<Sum>i\<in>P. f i x) \<in> borel_measurable M" "AE x in M. 0 \<le> (\<Sum>i\<in>P. f i x)"
1213       by (auto intro!: setsum_nonneg)
1214     from positive_integral_add[OF this]
1215     show ?case using insert by auto
1216   qed simp
1217 qed simp
1219 lemma positive_integral_Markov_inequality:
1220   assumes u: "u \<in> borel_measurable M" "AE x in M. 0 \<le> u x" and "A \<in> sets M" and c: "0 \<le> c"
1221   shows "(emeasure M) ({x\<in>space M. 1 \<le> c * u x} \<inter> A) \<le> c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
1222     (is "(emeasure M) ?A \<le> _ * ?PI")
1223 proof -
1224   have "?A \<in> sets M"
1225     using `A \<in> sets M` u by auto
1226   hence "(emeasure M) ?A = (\<integral>\<^isup>+ x. indicator ?A x \<partial>M)"
1227     using positive_integral_indicator by simp
1228   also have "\<dots> \<le> (\<integral>\<^isup>+ x. c * (u x * indicator A x) \<partial>M)" using u c
1229     by (auto intro!: positive_integral_mono_AE
1230       simp: indicator_def ereal_zero_le_0_iff)
1231   also have "\<dots> = c * (\<integral>\<^isup>+ x. u x * indicator A x \<partial>M)"
1232     using assms
1233     by (auto intro!: positive_integral_cmult simp: ereal_zero_le_0_iff)
1234   finally show ?thesis .
1235 qed
1237 lemma positive_integral_noteq_infinite:
1238   assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
1239   and "integral\<^isup>P M g \<noteq> \<infinity>"
1240   shows "AE x in M. g x \<noteq> \<infinity>"
1241 proof (rule ccontr)
1242   assume c: "\<not> (AE x in M. g x \<noteq> \<infinity>)"
1243   have "(emeasure M) {x\<in>space M. g x = \<infinity>} \<noteq> 0"
1244     using c g by (auto simp add: AE_iff_null)
1245   moreover have "0 \<le> (emeasure M) {x\<in>space M. g x = \<infinity>}" using g by (auto intro: measurable_sets)
1246   ultimately have "0 < (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto
1247   then have "\<infinity> = \<infinity> * (emeasure M) {x\<in>space M. g x = \<infinity>}" by auto
1248   also have "\<dots> \<le> (\<integral>\<^isup>+x. \<infinity> * indicator {x\<in>space M. g x = \<infinity>} x \<partial>M)"
1249     using g by (subst positive_integral_cmult_indicator) auto
1250   also have "\<dots> \<le> integral\<^isup>P M g"
1251     using assms by (auto intro!: positive_integral_mono_AE simp: indicator_def)
1252   finally show False using `integral\<^isup>P M g \<noteq> \<infinity>` by auto
1253 qed
1255 lemma positive_integral_diff:
1256   assumes f: "f \<in> borel_measurable M"
1257   and g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
1258   and fin: "integral\<^isup>P M g \<noteq> \<infinity>"
1259   and mono: "AE x in M. g x \<le> f x"
1260   shows "(\<integral>\<^isup>+ x. f x - g x \<partial>M) = integral\<^isup>P M f - integral\<^isup>P M g"
1261 proof -
1262   have diff: "(\<lambda>x. f x - g x) \<in> borel_measurable M" "AE x in M. 0 \<le> f x - g x"
1263     using assms by (auto intro: ereal_diff_positive)
1264   have pos_f: "AE x in M. 0 \<le> f x" using mono g by auto
1265   { fix a b :: ereal assume "0 \<le> a" "a \<noteq> \<infinity>" "0 \<le> b" "a \<le> b" then have "b - a + a = b"
1266       by (cases rule: ereal2_cases[of a b]) auto }
1267   note * = this
1268   then have "AE x in M. f x = f x - g x + g x"
1269     using mono positive_integral_noteq_infinite[OF g fin] assms by auto
1270   then have **: "integral\<^isup>P M f = (\<integral>\<^isup>+x. f x - g x \<partial>M) + integral\<^isup>P M g"
1271     unfolding positive_integral_add[OF diff g, symmetric]
1272     by (rule positive_integral_cong_AE)
1273   show ?thesis unfolding **
1274     using fin positive_integral_positive[of M g]
1275     by (cases rule: ereal2_cases[of "\<integral>\<^isup>+ x. f x - g x \<partial>M" "integral\<^isup>P M g"]) auto
1276 qed
1278 lemma positive_integral_suminf:
1279   assumes f: "\<And>i. f i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> f i x"
1280   shows "(\<integral>\<^isup>+ x. (\<Sum>i. f i x) \<partial>M) = (\<Sum>i. integral\<^isup>P M (f i))"
1281 proof -
1282   have all_pos: "AE x in M. \<forall>i. 0 \<le> f i x"
1283     using assms by (auto simp: AE_all_countable)
1284   have "(\<Sum>i. integral\<^isup>P M (f i)) = (SUP n. \<Sum>i<n. integral\<^isup>P M (f i))"
1285     using positive_integral_positive by (rule suminf_ereal_eq_SUPR)
1286   also have "\<dots> = (SUP n. \<integral>\<^isup>+x. (\<Sum>i<n. f i x) \<partial>M)"
1287     unfolding positive_integral_setsum[OF f] ..
1288   also have "\<dots> = \<integral>\<^isup>+x. (SUP n. \<Sum>i<n. f i x) \<partial>M" using f all_pos
1289     by (intro positive_integral_monotone_convergence_SUP_AE[symmetric])
1290        (elim AE_mp, auto simp: setsum_nonneg simp del: setsum_lessThan_Suc intro!: AE_I2 setsum_mono3)
1291   also have "\<dots> = \<integral>\<^isup>+x. (\<Sum>i. f i x) \<partial>M" using all_pos
1292     by (intro positive_integral_cong_AE) (auto simp: suminf_ereal_eq_SUPR)
1293   finally show ?thesis by simp
1294 qed
1296 text {* Fatou's lemma: convergence theorem on limes inferior *}
1297 lemma positive_integral_lim_INF:
1298   fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> ereal"
1299   assumes u: "\<And>i. u i \<in> borel_measurable M" "\<And>i. AE x in M. 0 \<le> u i x"
1300   shows "(\<integral>\<^isup>+ x. liminf (\<lambda>n. u n x) \<partial>M) \<le> liminf (\<lambda>n. integral\<^isup>P M (u n))"
1301 proof -
1302   have pos: "AE x in M. \<forall>i. 0 \<le> u i x" using u by (auto simp: AE_all_countable)
1303   have "(\<integral>\<^isup>+ x. liminf (\<lambda>n. u n x) \<partial>M) =
1304     (SUP n. \<integral>\<^isup>+ x. (INF i:{n..}. u i x) \<partial>M)"
1305     unfolding liminf_SUPR_INFI using pos u
1306     by (intro positive_integral_monotone_convergence_SUP_AE)
1307        (elim AE_mp, auto intro!: AE_I2 intro: INF_greatest INF_superset_mono)
1308   also have "\<dots> \<le> liminf (\<lambda>n. integral\<^isup>P M (u n))"
1309     unfolding liminf_SUPR_INFI
1310     by (auto intro!: SUP_mono exI INF_greatest positive_integral_mono INF_lower)
1311   finally show ?thesis .
1312 qed
1314 lemma positive_integral_null_set:
1315   assumes "N \<in> null_sets M" shows "(\<integral>\<^isup>+ x. u x * indicator N x \<partial>M) = 0"
1316 proof -
1317   have "(\<integral>\<^isup>+ x. u x * indicator N x \<partial>M) = (\<integral>\<^isup>+ x. 0 \<partial>M)"
1318   proof (intro positive_integral_cong_AE AE_I)
1319     show "{x \<in> space M. u x * indicator N x \<noteq> 0} \<subseteq> N"
1320       by (auto simp: indicator_def)
1321     show "(emeasure M) N = 0" "N \<in> sets M"
1322       using assms by auto
1323   qed
1324   then show ?thesis by simp
1325 qed
1327 lemma positive_integral_0_iff:
1328   assumes u: "u \<in> borel_measurable M" and pos: "AE x in M. 0 \<le> u x"
1329   shows "integral\<^isup>P M u = 0 \<longleftrightarrow> emeasure M {x\<in>space M. u x \<noteq> 0} = 0"
1330     (is "_ \<longleftrightarrow> (emeasure M) ?A = 0")
1331 proof -
1332   have u_eq: "(\<integral>\<^isup>+ x. u x * indicator ?A x \<partial>M) = integral\<^isup>P M u"
1333     by (auto intro!: positive_integral_cong simp: indicator_def)
1334   show ?thesis
1335   proof
1336     assume "(emeasure M) ?A = 0"
1337     with positive_integral_null_set[of ?A M u] u
1338     show "integral\<^isup>P M u = 0" by (simp add: u_eq null_sets_def)
1339   next
1340     { fix r :: ereal and n :: nat assume gt_1: "1 \<le> real n * r"
1341       then have "0 < real n * r" by (cases r) (auto split: split_if_asm simp: one_ereal_def)
1342       then have "0 \<le> r" by (auto simp add: ereal_zero_less_0_iff) }
1343     note gt_1 = this
1344     assume *: "integral\<^isup>P M u = 0"
1345     let ?M = "\<lambda>n. {x \<in> space M. 1 \<le> real (n::nat) * u x}"
1346     have "0 = (SUP n. (emeasure M) (?M n \<inter> ?A))"
1347     proof -
1348       { fix n :: nat
1349         from positive_integral_Markov_inequality[OF u pos, of ?A "ereal (real n)"]
1350         have "(emeasure M) (?M n \<inter> ?A) \<le> 0" unfolding u_eq * using u by simp
1351         moreover have "0 \<le> (emeasure M) (?M n \<inter> ?A)" using u by auto
1352         ultimately have "(emeasure M) (?M n \<inter> ?A) = 0" by auto }
1353       thus ?thesis by simp
1354     qed
1355     also have "\<dots> = (emeasure M) (\<Union>n. ?M n \<inter> ?A)"
1356     proof (safe intro!: SUP_emeasure_incseq)
1357       fix n show "?M n \<inter> ?A \<in> sets M"
1358         using u by (auto intro!: Int)
1359     next
1360       show "incseq (\<lambda>n. {x \<in> space M. 1 \<le> real n * u x} \<inter> {x \<in> space M. u x \<noteq> 0})"
1361       proof (safe intro!: incseq_SucI)
1362         fix n :: nat and x
1363         assume *: "1 \<le> real n * u x"
1364         also from gt_1[OF this] have "real n * u x \<le> real (Suc n) * u x"
1365           using `0 \<le> u x` by (auto intro!: ereal_mult_right_mono)
1366         finally show "1 \<le> real (Suc n) * u x" by auto
1367       qed
1368     qed
1369     also have "\<dots> = (emeasure M) {x\<in>space M. 0 < u x}"
1370     proof (safe intro!: arg_cong[where f="(emeasure M)"] dest!: gt_1)
1371       fix x assume "0 < u x" and [simp, intro]: "x \<in> space M"
1372       show "x \<in> (\<Union>n. ?M n \<inter> ?A)"
1373       proof (cases "u x")
1374         case (real r) with `0 < u x` have "0 < r" by auto
1375         obtain j :: nat where "1 / r \<le> real j" using real_arch_simple ..
1376         hence "1 / r * r \<le> real j * r" unfolding mult_le_cancel_right using `0 < r` by auto
1377         hence "1 \<le> real j * r" using real `0 < r` by auto
1378         thus ?thesis using `0 < r` real by (auto simp: one_ereal_def)
1379       qed (insert `0 < u x`, auto)
1380     qed auto
1381     finally have "(emeasure M) {x\<in>space M. 0 < u x} = 0" by simp
1382     moreover
1383     from pos have "AE x in M. \<not> (u x < 0)" by auto
1384     then have "(emeasure M) {x\<in>space M. u x < 0} = 0"
1385       using AE_iff_null[of M] u by auto
1386     moreover have "(emeasure M) {x\<in>space M. u x \<noteq> 0} = (emeasure M) {x\<in>space M. u x < 0} + (emeasure M) {x\<in>space M. 0 < u x}"
1387       using u by (subst plus_emeasure) (auto intro!: arg_cong[where f="emeasure M"])
1388     ultimately show "(emeasure M) ?A = 0" by simp
1389   qed
1390 qed
1392 lemma positive_integral_0_iff_AE:
1393   assumes u: "u \<in> borel_measurable M"
1394   shows "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x in M. u x \<le> 0)"
1395 proof -
1396   have sets: "{x\<in>space M. max 0 (u x) \<noteq> 0} \<in> sets M"
1397     using u by auto
1398   from positive_integral_0_iff[of "\<lambda>x. max 0 (u x)"]
1399   have "integral\<^isup>P M u = 0 \<longleftrightarrow> (AE x in M. max 0 (u x) = 0)"
1400     unfolding positive_integral_max_0
1401     using AE_iff_null[OF sets] u by auto
1402   also have "\<dots> \<longleftrightarrow> (AE x in M. u x \<le> 0)" by (auto split: split_max)
1403   finally show ?thesis .
1404 qed
1406 lemma AE_iff_positive_integral:
1407   "{x\<in>space M. P x} \<in> sets M \<Longrightarrow> (AE x in M. P x) \<longleftrightarrow> integral\<^isup>P M (indicator {x. \<not> P x}) = 0"
1408   by (subst positive_integral_0_iff_AE)
1409      (auto simp: one_ereal_def zero_ereal_def sets_Collect_neg indicator_def[abs_def] measurable_If)
1411 lemma positive_integral_const_If:
1412   "(\<integral>\<^isup>+x. a \<partial>M) = (if 0 \<le> a then a * (emeasure M) (space M) else 0)"
1413   by (auto intro!: positive_integral_0_iff_AE[THEN iffD2])
1415 lemma positive_integral_subalgebra:
1416   assumes f: "f \<in> borel_measurable N" "\<And>x. 0 \<le> f x"
1417   and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A"
1418   shows "integral\<^isup>P N f = integral\<^isup>P M f"
1419 proof -
1420   have [simp]: "\<And>f :: 'a \<Rightarrow> ereal. f \<in> borel_measurable N \<Longrightarrow> f \<in> borel_measurable M"
1421     using N by (auto simp: measurable_def)
1422   have [simp]: "\<And>P. (AE x in N. P x) \<Longrightarrow> (AE x in M. P x)"
1423     using N by (auto simp add: eventually_ae_filter null_sets_def)
1424   have [simp]: "\<And>A. A \<in> sets N \<Longrightarrow> A \<in> sets M"
1425     using N by auto
1426   from f show ?thesis
1427     apply induct
1428     apply (simp_all add: positive_integral_add positive_integral_cmult positive_integral_monotone_convergence_SUP N)
1429     apply (auto intro!: positive_integral_cong cong: positive_integral_cong simp: N(2)[symmetric])
1430     done
1431 qed
1433 section "Lebesgue Integral"
1435 definition integrable :: "'a measure \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool" where
1436   "integrable M f \<longleftrightarrow> f \<in> borel_measurable M \<and>
1437     (\<integral>\<^isup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity> \<and> (\<integral>\<^isup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
1439 lemma integrableD[dest]:
1440   assumes "integrable M f"
1441   shows "f \<in> borel_measurable M" "(\<integral>\<^isup>+ x. ereal (f x) \<partial>M) \<noteq> \<infinity>" "(\<integral>\<^isup>+ x. ereal (- f x) \<partial>M) \<noteq> \<infinity>"
1442   using assms unfolding integrable_def by auto
1444 definition lebesgue_integral :: "'a measure \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> real" ("integral\<^isup>L") where
1445   "integral\<^isup>L M f = real ((\<integral>\<^isup>+ x. ereal (f x) \<partial>M)) - real ((\<integral>\<^isup>+ x. ereal (- f x) \<partial>M))"
1447 syntax
1448   "_lebesgue_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> 'a measure \<Rightarrow> real" ("\<integral> _. _ \<partial>_" [60,61] 110)
1450 translations
1451   "\<integral> x. f \<partial>M" == "CONST lebesgue_integral M (%x. f)"
1453 lemma integrableE:
1454   assumes "integrable M f"
1455   obtains r q where
1456     "(\<integral>\<^isup>+x. ereal (f x)\<partial>M) = ereal r"
1457     "(\<integral>\<^isup>+x. ereal (-f x)\<partial>M) = ereal q"
1458     "f \<in> borel_measurable M" "integral\<^isup>L M f = r - q"
1459   using assms unfolding integrable_def lebesgue_integral_def
1460   using positive_integral_positive[of M "\<lambda>x. ereal (f x)"]
1461   using positive_integral_positive[of M "\<lambda>x. ereal (-f x)"]
1462   by (cases rule: ereal2_cases[of "(\<integral>\<^isup>+x. ereal (-f x)\<partial>M)" "(\<integral>\<^isup>+x. ereal (f x)\<partial>M)"]) auto
1464 lemma integral_cong:
1465   assumes "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
1466   shows "integral\<^isup>L M f = integral\<^isup>L M g"
1467   using assms by (simp cong: positive_integral_cong add: lebesgue_integral_def)
1469 lemma integral_cong_AE:
1470   assumes cong: "AE x in M. f x = g x"
1471   shows "integral\<^isup>L M f = integral\<^isup>L M g"
1472 proof -
1473   have *: "AE x in M. ereal (f x) = ereal (g x)"
1474     "AE x in M. ereal (- f x) = ereal (- g x)" using cong by auto
1475   show ?thesis
1476     unfolding *[THEN positive_integral_cong_AE] lebesgue_integral_def ..
1477 qed
1479 lemma integrable_cong_AE:
1480   assumes borel: "f \<in> borel_measurable M" "g \<in> borel_measurable M"
1481   assumes "AE x in M. f x = g x"
1482   shows "integrable M f = integrable M g"
1483 proof -
1484   have "(\<integral>\<^isup>+ x. ereal (f x) \<partial>M) = (\<integral>\<^isup>+ x. ereal (g x) \<partial>M)"
1485     "(\<integral>\<^isup>+ x. ereal (- f x) \<partial>M) = (\<integral>\<^isup>+ x. ereal (- g x) \<partial>M)"
1486     using assms by (auto intro!: positive_integral_cong_AE)
1487   with assms show ?thesis
1488     by (auto simp: integrable_def)
1489 qed
1491 lemma integrable_cong:
1492   "(\<And>x. x \<in> space M \<Longrightarrow> f x = g x) \<Longrightarrow> integrable M f \<longleftrightarrow> integrable M g"
1493   by (simp cong: positive_integral_cong measurable_cong add: integrable_def)
1495 lemma integral_mono_AE:
1496   assumes fg: "integrable M f" "integrable M g"
1497   and mono: "AE t in M. f t \<le> g t"
1498   shows "integral\<^isup>L M f \<le> integral\<^isup>L M g"
1499 proof -
1500   have "AE x in M. ereal (f x) \<le> ereal (g x)"
1501     using mono by auto
1502   moreover have "AE x in M. ereal (- g x) \<le> ereal (- f x)"
1503     using mono by auto
1504   ultimately show ?thesis using fg
1505     by (auto intro!: add_mono positive_integral_mono_AE real_of_ereal_positive_mono
1506              simp: positive_integral_positive lebesgue_integral_def diff_minus)
1507 qed
1509 lemma integral_mono:
1510   assumes "integrable M f" "integrable M g" "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
1511   shows "integral\<^isup>L M f \<le> integral\<^isup>L M g"
1512   using assms by (auto intro: integral_mono_AE)
1514 lemma positive_integral_eq_integral:
1515   assumes f: "integrable M f"
1516   assumes nonneg: "AE x in M. 0 \<le> f x"
1517   shows "(\<integral>\<^isup>+ x. ereal (f x) \<partial>M) = integral\<^isup>L M f"
1518 proof -
1519   have "(\<integral>\<^isup>+ x. max 0 (ereal (- f x)) \<partial>M) = (\<integral>\<^isup>+ x. 0 \<partial>M)"
1520     using nonneg by (intro positive_integral_cong_AE) (auto split: split_max)
1521   with f positive_integral_positive show ?thesis
1522     by (cases "\<integral>\<^isup>+ x. ereal (f x) \<partial>M")
1523        (auto simp add: lebesgue_integral_def positive_integral_max_0 integrable_def)
1524 qed
1526 lemma integral_eq_positive_integral:
1527   assumes f: "\<And>x. 0 \<le> f x"
1528   shows "integral\<^isup>L M f = real (\<integral>\<^isup>+ x. ereal (f x) \<partial>M)"
1529 proof -
1530   { fix x have "max 0 (ereal (- f x)) = 0" using f[of x] by (simp split: split_max) }
1531   then have "0 = (\<integral>\<^isup>+ x. max 0 (ereal (- f x)) \<partial>M)" by simp
1532   also have "\<dots> = (\<integral>\<^isup>+ x. ereal (- f x) \<partial>M)" unfolding positive_integral_max_0 ..
1533   finally show ?thesis
1534     unfolding lebesgue_integral_def by simp
1535 qed
1537 lemma integral_minus[intro, simp]:
1538   assumes "integrable M f"
1539   shows "integrable M (\<lambda>x. - f x)" "(\<integral>x. - f x \<partial>M) = - integral\<^isup>L M f"
1540   using assms by (auto simp: integrable_def lebesgue_integral_def)
1542 lemma integral_minus_iff[simp]:
1543   "integrable M (\<lambda>x. - f x) \<longleftrightarrow> integrable M f"
1544 proof
1545   assume "integrable M (\<lambda>x. - f x)"
1546   then have "integrable M (\<lambda>x. - (- f x))"
1547     by (rule integral_minus)
1548   then show "integrable M f" by simp
1549 qed (rule integral_minus)
1551 lemma integral_of_positive_diff:
1552   assumes integrable: "integrable M u" "integrable M v"
1553   and f_def: "\<And>x. f x = u x - v x" and pos: "\<And>x. 0 \<le> u x" "\<And>x. 0 \<le> v x"
1554   shows "integrable M f" and "integral\<^isup>L M f = integral\<^isup>L M u - integral\<^isup>L M v"
1555 proof -
1556   let ?f = "\<lambda>x. max 0 (ereal (f x))"
1557   let ?mf = "\<lambda>x. max 0 (ereal (- f x))"
1558   let ?u = "\<lambda>x. max 0 (ereal (u x))"
1559   let ?v = "\<lambda>x. max 0 (ereal (v x))"
1561   from borel_measurable_diff[of u M v] integrable
1562   have f_borel: "?f \<in> borel_measurable M" and
1563     mf_borel: "?mf \<in> borel_measurable M" and
1564     v_borel: "?v \<in> borel_measurable M" and
1565     u_borel: "?u \<in> borel_measurable M" and
1566     "f \<in> borel_measurable M"
1567     by (auto simp: f_def[symmetric] integrable_def)
1569   have "(\<integral>\<^isup>+ x. ereal (u x - v x) \<partial>M) \<le> integral\<^isup>P M ?u"
1570     using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
1571   moreover have "(\<integral>\<^isup>+ x. ereal (v x - u x) \<partial>M) \<le> integral\<^isup>P M ?v"
1572     using pos by (auto intro!: positive_integral_mono simp: positive_integral_max_0)
1573   ultimately show f: "integrable M f"
1574     using `integrable M u` `integrable M v` `f \<in> borel_measurable M`
1575     by (auto simp: integrable_def f_def positive_integral_max_0)
1577   have "\<And>x. ?u x + ?mf x = ?v x + ?f x"
1578     unfolding f_def using pos by (simp split: split_max)
1579   then have "(\<integral>\<^isup>+ x. ?u x + ?mf x \<partial>M) = (\<integral>\<^isup>+ x. ?v x + ?f x \<partial>M)" by simp
1580   then have "real (integral\<^isup>P M ?u + integral\<^isup>P M ?mf) =
1581       real (integral\<^isup>P M ?v + integral\<^isup>P M ?f)"
1582     using positive_integral_add[OF u_borel _ mf_borel]
1583     using positive_integral_add[OF v_borel _ f_borel]
1584     by auto
1585   then show "integral\<^isup>L M f = integral\<^isup>L M u - integral\<^isup>L M v"
1586     unfolding positive_integral_max_0
1587     unfolding pos[THEN integral_eq_positive_integral]
1588     using integrable f by (auto elim!: integrableE)
1589 qed
1591 lemma integral_linear:
1592   assumes "integrable M f" "integrable M g" and "0 \<le> a"
1593   shows "integrable M (\<lambda>t. a * f t + g t)"
1594   and "(\<integral> t. a * f t + g t \<partial>M) = a * integral\<^isup>L M f + integral\<^isup>L M g" (is ?EQ)
1595 proof -
1596   let ?f = "\<lambda>x. max 0 (ereal (f x))"
1597   let ?g = "\<lambda>x. max 0 (ereal (g x))"
1598   let ?mf = "\<lambda>x. max 0 (ereal (- f x))"
1599   let ?mg = "\<lambda>x. max 0 (ereal (- g x))"
1600   let ?p = "\<lambda>t. max 0 (a * f t) + max 0 (g t)"
1601   let ?n = "\<lambda>t. max 0 (- (a * f t)) + max 0 (- g t)"
1603   from assms have linear:
1604     "(\<integral>\<^isup>+ x. ereal a * ?f x + ?g x \<partial>M) = ereal a * integral\<^isup>P M ?f + integral\<^isup>P M ?g"
1605     "(\<integral>\<^isup>+ x. ereal a * ?mf x + ?mg x \<partial>M) = ereal a * integral\<^isup>P M ?mf + integral\<^isup>P M ?mg"
1606     by (auto intro!: positive_integral_linear simp: integrable_def)
1608   have *: "(\<integral>\<^isup>+x. ereal (- ?p x) \<partial>M) = 0" "(\<integral>\<^isup>+x. ereal (- ?n x) \<partial>M) = 0"
1609     using `0 \<le> a` assms by (auto simp: positive_integral_0_iff_AE integrable_def)
1610   have **: "\<And>x. ereal a * ?f x + ?g x = max 0 (ereal (?p x))"
1611            "\<And>x. ereal a * ?mf x + ?mg x = max 0 (ereal (?n x))"
1612     using `0 \<le> a` by (auto split: split_max simp: zero_le_mult_iff mult_le_0_iff)
1614   have "integrable M ?p" "integrable M ?n"
1615       "\<And>t. a * f t + g t = ?p t - ?n t" "\<And>t. 0 \<le> ?p t" "\<And>t. 0 \<le> ?n t"
1616     using linear assms unfolding integrable_def ** *
1617     by (auto simp: positive_integral_max_0)
1618   note diff = integral_of_positive_diff[OF this]
1620   show "integrable M (\<lambda>t. a * f t + g t)" by (rule diff)
1621   from assms linear show ?EQ
1622     unfolding diff(2) ** positive_integral_max_0
1623     unfolding lebesgue_integral_def *
1624     by (auto elim!: integrableE simp: field_simps)
1625 qed
1627 lemma integral_add[simp, intro]:
1628   assumes "integrable M f" "integrable M g"
1629   shows "integrable M (\<lambda>t. f t + g t)"
1630   and "(\<integral> t. f t + g t \<partial>M) = integral\<^isup>L M f + integral\<^isup>L M g"
1631   using assms integral_linear[where a=1] by auto
1633 lemma integral_zero[simp, intro]:
1634   shows "integrable M (\<lambda>x. 0)" "(\<integral> x.0 \<partial>M) = 0"
1635   unfolding integrable_def lebesgue_integral_def
1636   by auto
1638 lemma integral_cmult[simp, intro]:
1639   assumes "integrable M f"
1640   shows "integrable M (\<lambda>t. a * f t)" (is ?P)
1641   and "(\<integral> t. a * f t \<partial>M) = a * integral\<^isup>L M f" (is ?I)
1642 proof -
1643   have "integrable M (\<lambda>t. a * f t) \<and> (\<integral> t. a * f t \<partial>M) = a * integral\<^isup>L M f"
1644   proof (cases rule: le_cases)
1645     assume "0 \<le> a" show ?thesis
1646       using integral_linear[OF assms integral_zero(1) `0 \<le> a`]
1647       by simp
1648   next
1649     assume "a \<le> 0" hence "0 \<le> - a" by auto
1650     have *: "\<And>t. - a * t + 0 = (-a) * t" by simp
1651     show ?thesis using integral_linear[OF assms integral_zero(1) `0 \<le> - a`]
1652         integral_minus(1)[of M "\<lambda>t. - a * f t"]
1653       unfolding * integral_zero by simp
1654   qed
1655   thus ?P ?I by auto
1656 qed
1658 lemma lebesgue_integral_cmult_nonneg:
1659   assumes f: "f \<in> borel_measurable M" and "0 \<le> c"
1660   shows "(\<integral>x. c * f x \<partial>M) = c * integral\<^isup>L M f"
1661 proof -
1662   { have "real (ereal c * integral\<^isup>P M (\<lambda>x. max 0 (ereal (f x)))) =
1663       real (integral\<^isup>P M (\<lambda>x. ereal c * max 0 (ereal (f x))))"
1664       using f `0 \<le> c` by (subst positive_integral_cmult) auto
1665     also have "\<dots> = real (integral\<^isup>P M (\<lambda>x. max 0 (ereal (c * f x))))"
1666       using `0 \<le> c` by (auto intro!: arg_cong[where f=real] positive_integral_cong simp: max_def zero_le_mult_iff)
1667     finally have "real (integral\<^isup>P M (\<lambda>x. ereal (c * f x))) = c * real (integral\<^isup>P M (\<lambda>x. ereal (f x)))"
1668       by (simp add: positive_integral_max_0) }
1669   moreover
1670   { have "real (ereal c * integral\<^isup>P M (\<lambda>x. max 0 (ereal (- f x)))) =
1671       real (integral\<^isup>P M (\<lambda>x. ereal c * max 0 (ereal (- f x))))"
1672       using f `0 \<le> c` by (subst positive_integral_cmult) auto
1673     also have "\<dots> = real (integral\<^isup>P M (\<lambda>x. max 0 (ereal (- c * f x))))"
1674       using `0 \<le> c` by (auto intro!: arg_cong[where f=real] positive_integral_cong simp: max_def mult_le_0_iff)
1675     finally have "real (integral\<^isup>P M (\<lambda>x. ereal (- c * f x))) = c * real (integral\<^isup>P M (\<lambda>x. ereal (- f x)))"
1676       by (simp add: positive_integral_max_0) }
1677   ultimately show ?thesis
1678     by (simp add: lebesgue_integral_def field_simps)
1679 qed
1681 lemma lebesgue_integral_uminus:
1682   "(\<integral>x. - f x \<partial>M) = - integral\<^isup>L M f"
1683     unfolding lebesgue_integral_def by simp
1685 lemma lebesgue_integral_cmult:
1686   assumes f: "f \<in> borel_measurable M"
1687   shows "(\<integral>x. c * f x \<partial>M) = c * integral\<^isup>L M f"
1688 proof (cases rule: linorder_le_cases)
1689   assume "0 \<le> c" with f show ?thesis by (rule lebesgue_integral_cmult_nonneg)
1690 next
1691   assume "c \<le> 0"
1692   with lebesgue_integral_cmult_nonneg[OF f, of "-c"]
1693   show ?thesis
1694     by (simp add: lebesgue_integral_def)
1695 qed
1697 lemma integral_multc:
1698   assumes "integrable M f"
1699   shows "(\<integral> x. f x * c \<partial>M) = integral\<^isup>L M f * c"
1700   unfolding mult_commute[of _ c] integral_cmult[OF assms] ..
1702 lemma integral_diff[simp, intro]:
1703   assumes f: "integrable M f" and g: "integrable M g"
1704   shows "integrable M (\<lambda>t. f t - g t)"
1705   and "(\<integral> t. f t - g t \<partial>M) = integral\<^isup>L M f - integral\<^isup>L M g"
1706   using integral_add[OF f integral_minus(1)[OF g]]
1707   unfolding diff_minus integral_minus(2)[OF g]
1708   by auto
1710 lemma integral_indicator[simp, intro]:
1711   assumes "A \<in> sets M" and "(emeasure M) A \<noteq> \<infinity>"
1712   shows "integral\<^isup>L M (indicator A) = real ((emeasure M) A)" (is ?int)
1713   and "integrable M (indicator A)" (is ?able)
1714 proof -
1715   from `A \<in> sets M` have *:
1716     "\<And>x. ereal (indicator A x) = indicator A x"
1717     "(\<integral>\<^isup>+x. ereal (- indicator A x) \<partial>M) = 0"
1718     by (auto split: split_indicator simp: positive_integral_0_iff_AE one_ereal_def)
1719   show ?int ?able
1720     using assms unfolding lebesgue_integral_def integrable_def
1721     by (auto simp: *)
1722 qed
1724 lemma integral_cmul_indicator:
1725   assumes "A \<in> sets M" and "c \<noteq> 0 \<Longrightarrow> (emeasure M) A \<noteq> \<infinity>"
1726   shows "integrable M (\<lambda>x. c * indicator A x)" (is ?P)
1727   and "(\<integral>x. c * indicator A x \<partial>M) = c * real ((emeasure M) A)" (is ?I)
1728 proof -
1729   show ?P
1730   proof (cases "c = 0")
1731     case False with assms show ?thesis by simp
1732   qed simp
1734   show ?I
1735   proof (cases "c = 0")
1736     case False with assms show ?thesis by simp
1737   qed simp
1738 qed
1740 lemma integral_setsum[simp, intro]:
1741   assumes "\<And>n. n \<in> S \<Longrightarrow> integrable M (f n)"
1742   shows "(\<integral>x. (\<Sum> i \<in> S. f i x) \<partial>M) = (\<Sum> i \<in> S. integral\<^isup>L M (f i))" (is "?int S")
1743     and "integrable M (\<lambda>x. \<Sum> i \<in> S. f i x)" (is "?I S")
1744 proof -
1745   have "?int S \<and> ?I S"
1746   proof (cases "finite S")
1747     assume "finite S"
1748     from this assms show ?thesis by (induct S) simp_all
1749   qed simp
1750   thus "?int S" and "?I S" by auto
1751 qed
1753 lemma integrable_bound:
1754   assumes "integrable M f" and f: "AE x in M. \<bar>g x\<bar> \<le> f x"
1755   assumes borel: "g \<in> borel_measurable M"
1756   shows "integrable M g"
1757 proof -
1758   have "(\<integral>\<^isup>+ x. ereal (g x) \<partial>M) \<le> (\<integral>\<^isup>+ x. ereal \<bar>g x\<bar> \<partial>M)"
1759     by (auto intro!: positive_integral_mono)
1760   also have "\<dots> \<le> (\<integral>\<^isup>+ x. ereal (f x) \<partial>M)"
1761     using f by (auto intro!: positive_integral_mono_AE)
1762   also have "\<dots> < \<infinity>"
1763     using `integrable M f` unfolding integrable_def by auto
1764   finally have pos: "(\<integral>\<^isup>+ x. ereal (g x) \<partial>M) < \<infinity>" .
1766   have "(\<integral>\<^isup>+ x. ereal (- g x) \<partial>M) \<le> (\<integral>\<^isup>+ x. ereal (\<bar>g x\<bar>) \<partial>M)"
1767     by (auto intro!: positive_integral_mono)
1768   also have "\<dots> \<le> (\<integral>\<^isup>+ x. ereal (f x) \<partial>M)"
1769     using f by (auto intro!: positive_integral_mono_AE)
1770   also have "\<dots> < \<infinity>"
1771     using `integrable M f` unfolding integrable_def by auto
1772   finally have neg: "(\<integral>\<^isup>+ x. ereal (- g x) \<partial>M) < \<infinity>" .
1774   from neg pos borel show ?thesis
1775     unfolding integrable_def by auto
1776 qed
1778 lemma integrable_abs:
1779   assumes f: "integrable M f"
1780   shows "integrable M (\<lambda> x. \<bar>f x\<bar>)"
1781 proof -
1782   from assms have *: "(\<integral>\<^isup>+x. ereal (- \<bar>f x\<bar>)\<partial>M) = 0"
1783     "\<And>x. ereal \<bar>f x\<bar> = max 0 (ereal (f x)) + max 0 (ereal (- f x))"
1784     by (auto simp: integrable_def positive_integral_0_iff_AE split: split_max)
1785   with assms show ?thesis
1786     by (simp add: positive_integral_add positive_integral_max_0 integrable_def)
1787 qed
1789 lemma integral_subalgebra:
1790   assumes borel: "f \<in> borel_measurable N"
1791   and N: "sets N \<subseteq> sets M" "space N = space M" "\<And>A. A \<in> sets N \<Longrightarrow> emeasure N A = emeasure M A"
1792   shows "integrable N f \<longleftrightarrow> integrable M f" (is ?P)
1793     and "integral\<^isup>L N f = integral\<^isup>L M f" (is ?I)
1794 proof -
1795   have "(\<integral>\<^isup>+ x. max 0 (ereal (f x)) \<partial>N) = (\<integral>\<^isup>+ x. max 0 (ereal (f x)) \<partial>M)"
1796        "(\<integral>\<^isup>+ x. max 0 (ereal (- f x)) \<partial>N) = (\<integral>\<^isup>+ x. max 0 (ereal (- f x)) \<partial>M)"
1797     using borel by (auto intro!: positive_integral_subalgebra N)
1798   moreover have "f \<in> borel_measurable M \<longleftrightarrow> f \<in> borel_measurable N"
1799     using assms unfolding measurable_def by auto
1800   ultimately show ?P ?I
1801     by (auto simp: integrable_def lebesgue_integral_def positive_integral_max_0)
1802 qed
1804 lemma lebesgue_integral_nonneg:
1805   assumes ae: "(AE x in M. 0 \<le> f x)" shows "0 \<le> integral\<^isup>L M f"
1806 proof -
1807   have "(\<integral>\<^isup>+x. max 0 (ereal (- f x)) \<partial>M) = (\<integral>\<^isup>+x. 0 \<partial>M)"
1808     using ae by (intro positive_integral_cong_AE) (auto simp: max_def)
1809   then show ?thesis
1810     by (auto simp: lebesgue_integral_def positive_integral_max_0
1811              intro!: real_of_ereal_pos positive_integral_positive)
1812 qed
1814 lemma integrable_abs_iff:
1815   "f \<in> borel_measurable M \<Longrightarrow> integrable M (\<lambda> x. \<bar>f x\<bar>) \<longleftrightarrow> integrable M f"
1816   by (auto intro!: integrable_bound[where g=f] integrable_abs)
1818 lemma integrable_max:
1819   assumes int: "integrable M f" "integrable M g"
1820   shows "integrable M (\<lambda> x. max (f x) (g x))"
1821 proof (rule integrable_bound)
1822   show "integrable M (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
1823     using int by (simp add: integrable_abs)
1824   show "(\<lambda>x. max (f x) (g x)) \<in> borel_measurable M"
1825     using int unfolding integrable_def by auto
1826 qed auto
1828 lemma integrable_min:
1829   assumes int: "integrable M f" "integrable M g"
1830   shows "integrable M (\<lambda> x. min (f x) (g x))"
1831 proof (rule integrable_bound)
1832   show "integrable M (\<lambda>x. \<bar>f x\<bar> + \<bar>g x\<bar>)"
1833     using int by (simp add: integrable_abs)
1834   show "(\<lambda>x. min (f x) (g x)) \<in> borel_measurable M"
1835     using int unfolding integrable_def by auto
1836 qed auto
1838 lemma integral_triangle_inequality:
1839   assumes "integrable M f"
1840   shows "\<bar>integral\<^isup>L M f\<bar> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
1841 proof -
1842   have "\<bar>integral\<^isup>L M f\<bar> = max (integral\<^isup>L M f) (- integral\<^isup>L M f)" by auto
1843   also have "\<dots> \<le> (\<integral>x. \<bar>f x\<bar> \<partial>M)"
1844       using assms integral_minus(2)[of M f, symmetric]
1845       by (auto intro!: integral_mono integrable_abs simp del: integral_minus)
1846   finally show ?thesis .
1847 qed
1849 lemma integral_positive:
1850   assumes "integrable M f" "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"
1851   shows "0 \<le> integral\<^isup>L M f"
1852 proof -
1853   have "0 = (\<integral>x. 0 \<partial>M)" by auto
1854   also have "\<dots> \<le> integral\<^isup>L M f"
1855     using assms by (rule integral_mono[OF integral_zero(1)])
1856   finally show ?thesis .
1857 qed
1859 lemma integral_monotone_convergence_pos:
1860   assumes i: "\<And>i. integrable M (f i)" and mono: "AE x in M. mono (\<lambda>n. f n x)"
1861     and pos: "\<And>i. AE x in M. 0 \<le> f i x"
1862     and lim: "AE x in M. (\<lambda>i. f i x) ----> u x"
1863     and ilim: "(\<lambda>i. integral\<^isup>L M (f i)) ----> x"
1864     and u: "u \<in> borel_measurable M"
1865   shows "integrable M u"
1866   and "integral\<^isup>L M u = x"
1867 proof -
1868   have "(\<integral>\<^isup>+ x. ereal (u x) \<partial>M) = (SUP n. (\<integral>\<^isup>+ x. ereal (f n x) \<partial>M))"
1869   proof (subst positive_integral_monotone_convergence_SUP_AE[symmetric])
1870     fix i
1871     from mono pos show "AE x in M. ereal (f i x) \<le> ereal (f (Suc i) x) \<and> 0 \<le> ereal (f i x)"
1872       by eventually_elim (auto simp: mono_def)
1873     show "(\<lambda>x. ereal (f i x)) \<in> borel_measurable M"
1874       using i by (auto simp: integrable_def)
1875   next
1876     show "(\<integral>\<^isup>+ x. ereal (u x) \<partial>M) = \<integral>\<^isup>+ x. (SUP i. ereal (f i x)) \<partial>M"
1877       apply (rule positive_integral_cong_AE)
1878       using lim mono
1879       by eventually_elim (simp add: SUP_eq_LIMSEQ[THEN iffD2])
1880   qed
1881   also have "\<dots> = ereal x"
1882     using mono i unfolding positive_integral_eq_integral[OF i pos]
1883     by (subst SUP_eq_LIMSEQ) (auto simp: mono_def intro!: integral_mono_AE ilim)
1884   finally have "(\<integral>\<^isup>+ x. ereal (u x) \<partial>M) = ereal x" .
1885   moreover have "(\<integral>\<^isup>+ x. ereal (- u x) \<partial>M) = 0"
1886   proof (subst positive_integral_0_iff_AE)
1887     show "(\<lambda>x. ereal (- u x)) \<in> borel_measurable M"
1888       using u by auto
1889     from mono pos[of 0] lim show "AE x in M. ereal (- u x) \<le> 0"
1890     proof eventually_elim
1891       fix x assume "mono (\<lambda>n. f n x)" "0 \<le> f 0 x" "(\<lambda>i. f i x) ----> u x"
1892       then show "ereal (- u x) \<le> 0"
1893         using incseq_le[of "\<lambda>n. f n x" "u x" 0] by (simp add: mono_def incseq_def)
1894     qed
1895   qed
1896   ultimately show "integrable M u" "integral\<^isup>L M u = x"
1897     by (auto simp: integrable_def lebesgue_integral_def u)
1898 qed
1900 lemma integral_monotone_convergence:
1901   assumes f: "\<And>i. integrable M (f i)" and mono: "AE x in M. mono (\<lambda>n. f n x)"
1902   and lim: "AE x in M. (\<lambda>i. f i x) ----> u x"
1903   and ilim: "(\<lambda>i. integral\<^isup>L M (f i)) ----> x"
1904   and u: "u \<in> borel_measurable M"
1905   shows "integrable M u"
1906   and "integral\<^isup>L M u = x"
1907 proof -
1908   have 1: "\<And>i. integrable M (\<lambda>x. f i x - f 0 x)"
1909     using f by auto
1910   have 2: "AE x in M. mono (\<lambda>n. f n x - f 0 x)"
1911     using mono by (auto simp: mono_def le_fun_def)
1912   have 3: "\<And>n. AE x in M. 0 \<le> f n x - f 0 x"
1913     using mono by (auto simp: field_simps mono_def le_fun_def)
1914   have 4: "AE x in M. (\<lambda>i. f i x - f 0 x) ----> u x - f 0 x"
1915     using lim by (auto intro!: tendsto_diff)
1916   have 5: "(\<lambda>i. (\<integral>x. f i x - f 0 x \<partial>M)) ----> x - integral\<^isup>L M (f 0)"
1917     using f ilim by (auto intro!: tendsto_diff)
1918   have 6: "(\<lambda>x. u x - f 0 x) \<in> borel_measurable M"
1919     using f[of 0] u by auto
1920   note diff = integral_monotone_convergence_pos[OF 1 2 3 4 5 6]
1921   have "integrable M (\<lambda>x. (u x - f 0 x) + f 0 x)"
1922     using diff(1) f by (rule integral_add(1))
1923   with diff(2) f show "integrable M u" "integral\<^isup>L M u = x"
1924     by auto
1925 qed
1927 lemma integral_0_iff:
1928   assumes "integrable M f"
1929   shows "(\<integral>x. \<bar>f x\<bar> \<partial>M) = 0 \<longleftrightarrow> (emeasure M) {x\<in>space M. f x \<noteq> 0} = 0"
1930 proof -
1931   have *: "(\<integral>\<^isup>+x. ereal (- \<bar>f x\<bar>) \<partial>M) = 0"
1932     using assms by (auto simp: positive_integral_0_iff_AE integrable_def)
1933   have "integrable M (\<lambda>x. \<bar>f x\<bar>)" using assms by (rule integrable_abs)
1934   hence "(\<lambda>x. ereal (\<bar>f x\<bar>)) \<in> borel_measurable M"
1935     "(\<integral>\<^isup>+ x. ereal \<bar>f x\<bar> \<partial>M) \<noteq> \<infinity>" unfolding integrable_def by auto
1936   from positive_integral_0_iff[OF this(1)] this(2)
1937   show ?thesis unfolding lebesgue_integral_def *
1938     using positive_integral_positive[of M "\<lambda>x. ereal \<bar>f x\<bar>"]
1939     by (auto simp add: real_of_ereal_eq_0)
1940 qed
1942 lemma positive_integral_PInf:
1943   assumes f: "f \<in> borel_measurable M"
1944   and not_Inf: "integral\<^isup>P M f \<noteq> \<infinity>"
1945   shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
1946 proof -
1947   have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) = (\<integral>\<^isup>+ x. \<infinity> * indicator (f -` {\<infinity>} \<inter> space M) x \<partial>M)"
1948     using f by (subst positive_integral_cmult_indicator) (auto simp: measurable_sets)
1949   also have "\<dots> \<le> integral\<^isup>P M (\<lambda>x. max 0 (f x))"
1950     by (auto intro!: positive_integral_mono simp: indicator_def max_def)
1951   finally have "\<infinity> * (emeasure M) (f -` {\<infinity>} \<inter> space M) \<le> integral\<^isup>P M f"
1952     by (simp add: positive_integral_max_0)
1953   moreover have "0 \<le> (emeasure M) (f -` {\<infinity>} \<inter> space M)"
1954     by (rule emeasure_nonneg)
1955   ultimately show ?thesis
1956     using assms by (auto split: split_if_asm)
1957 qed
1959 lemma positive_integral_PInf_AE:
1960   assumes "f \<in> borel_measurable M" "integral\<^isup>P M f \<noteq> \<infinity>" shows "AE x in M. f x \<noteq> \<infinity>"
1961 proof (rule AE_I)
1962   show "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
1963     by (rule positive_integral_PInf[OF assms])
1964   show "f -` {\<infinity>} \<inter> space M \<in> sets M"
1965     using assms by (auto intro: borel_measurable_vimage)
1966 qed auto
1968 lemma simple_integral_PInf:
1969   assumes "simple_function M f" "\<And>x. 0 \<le> f x"
1970   and "integral\<^isup>S M f \<noteq> \<infinity>"
1971   shows "(emeasure M) (f -` {\<infinity>} \<inter> space M) = 0"
1972 proof (rule positive_integral_PInf)
1973   show "f \<in> borel_measurable M" using assms by (auto intro: borel_measurable_simple_function)
1974   show "integral\<^isup>P M f \<noteq> \<infinity>"
1975     using assms by (simp add: positive_integral_eq_simple_integral)
1976 qed
1978 lemma integral_real:
1979   "AE x in M. \<bar>f x\<bar> \<noteq> \<infinity> \<Longrightarrow> (\<integral>x. real (f x) \<partial>M) = real (integral\<^isup>P M f) - real (integral\<^isup>P M (\<lambda>x. - f x))"
1980   using assms unfolding lebesgue_integral_def
1981   by (subst (1 2) positive_integral_cong_AE) (auto simp add: ereal_real)
1983 lemma (in finite_measure) lebesgue_integral_const[simp]:
1984   shows "integrable M (\<lambda>x. a)"
1985   and  "(\<integral>x. a \<partial>M) = a * (measure M) (space M)"
1986 proof -
1987   { fix a :: real assume "0 \<le> a"
1988     then have "(\<integral>\<^isup>+ x. ereal a \<partial>M) = ereal a * (emeasure M) (space M)"
1989       by (subst positive_integral_const) auto
1990     moreover
1991     from `0 \<le> a` have "(\<integral>\<^isup>+ x. ereal (-a) \<partial>M) = 0"
1992       by (subst positive_integral_0_iff_AE) auto
1993     ultimately have "integrable M (\<lambda>x. a)" by (auto simp: integrable_def) }
1994   note * = this
1995   show "integrable M (\<lambda>x. a)"
1996   proof cases
1997     assume "0 \<le> a" with * show ?thesis .
1998   next
1999     assume "\<not> 0 \<le> a"
2000     then have "0 \<le> -a" by auto
2001     from *[OF this] show ?thesis by simp
2002   qed
2003   show "(\<integral>x. a \<partial>M) = a * measure M (space M)"
2004     by (simp add: lebesgue_integral_def positive_integral_const_If emeasure_eq_measure)
2005 qed
2007 lemma indicator_less[simp]:
2008   "indicator A x \<le> (indicator B x::ereal) \<longleftrightarrow> (x \<in> A \<longrightarrow> x \<in> B)"
2009   by (simp add: indicator_def not_le)
2011 lemma (in finite_measure) integral_less_AE:
2012   assumes int: "integrable M X" "integrable M Y"
2013   assumes A: "(emeasure M) A \<noteq> 0" "A \<in> sets M" "AE x in M. x \<in> A \<longrightarrow> X x \<noteq> Y x"
2014   assumes gt: "AE x in M. X x \<le> Y x"
2015   shows "integral\<^isup>L M X < integral\<^isup>L M Y"
2016 proof -
2017   have "integral\<^isup>L M X \<le> integral\<^isup>L M Y"
2018     using gt int by (intro integral_mono_AE) auto
2019   moreover
2020   have "integral\<^isup>L M X \<noteq> integral\<^isup>L M Y"
2021   proof
2022     assume eq: "integral\<^isup>L M X = integral\<^isup>L M Y"
2023     have "integral\<^isup>L M (\<lambda>x. \<bar>Y x - X x\<bar>) = integral\<^isup>L M (\<lambda>x. Y x - X x)"
2024       using gt by (intro integral_cong_AE) auto
2025     also have "\<dots> = 0"
2026       using eq int by simp
2027     finally have "(emeasure M) {x \<in> space M. Y x - X x \<noteq> 0} = 0"
2028       using int by (simp add: integral_0_iff)
2029     moreover
2030     have "(\<integral>\<^isup>+x. indicator A x \<partial>M) \<le> (\<integral>\<^isup>+x. indicator {x \<in> space M. Y x - X x \<noteq> 0} x \<partial>M)"
2031       using A by (intro positive_integral_mono_AE) auto
2032     then have "(emeasure M) A \<le> (emeasure M) {x \<in> space M. Y x - X x \<noteq> 0}"
2033       using int A by (simp add: integrable_def)
2034     ultimately have "emeasure M A = 0"
2035       using emeasure_nonneg[of M A] by simp
2036     with `(emeasure M) A \<noteq> 0` show False by auto
2037   qed
2038   ultimately show ?thesis by auto
2039 qed
2041 lemma (in finite_measure) integral_less_AE_space:
2042   assumes int: "integrable M X" "integrable M Y"
2043   assumes gt: "AE x in M. X x < Y x" "(emeasure M) (space M) \<noteq> 0"
2044   shows "integral\<^isup>L M X < integral\<^isup>L M Y"
2045   using gt by (intro integral_less_AE[OF int, where A="space M"]) auto
2047 lemma integral_dominated_convergence:
2048   assumes u: "\<And>i. integrable M (u i)" and bound: "\<And>j. AE x in M. \<bar>u j x\<bar> \<le> w x"
2049   and w: "integrable M w"
2050   and u': "AE x in M. (\<lambda>i. u i x) ----> u' x"
2051   and borel: "u' \<in> borel_measurable M"
2052   shows "integrable M u'"
2053   and "(\<lambda>i. (\<integral>x. \<bar>u i x - u' x\<bar> \<partial>M)) ----> 0" (is "?lim_diff")
2054   and "(\<lambda>i. integral\<^isup>L M (u i)) ----> integral\<^isup>L M u'" (is ?lim)
2055 proof -
2056   have all_bound: "AE x in M. \<forall>j. \<bar>u j x\<bar> \<le> w x"
2057     using bound by (auto simp: AE_all_countable)
2058   with u' have u'_bound: "AE x in M. \<bar>u' x\<bar> \<le> w x"
2059     by eventually_elim (auto intro: LIMSEQ_le_const2 tendsto_rabs)
2061   from bound[of 0] have w_pos: "AE x in M. 0 \<le> w x"
2062     by eventually_elim auto
2064   show "integrable M u'"
2065     by (rule integrable_bound) fact+
2067   let ?diff = "\<lambda>n x. 2 * w x - \<bar>u n x - u' x\<bar>"
2068   have diff: "\<And>n. integrable M (\<lambda>x. \<bar>u n x - u' x\<bar>)"
2069     using w u `integrable M u'` by (auto intro!: integrable_abs)
2071   from u'_bound all_bound
2072   have diff_less_2w: "AE x in M. \<forall>j. \<bar>u j x - u' x\<bar> \<le> 2 * w x"
2073   proof (eventually_elim, intro allI)
2074     fix x j assume *: "\<bar>u' x\<bar> \<le> w x" "\<forall>j. \<bar>u j x\<bar> \<le> w x"
2075     then have "\<bar>u j x - u' x\<bar> \<le> \<bar>u j x\<bar> + \<bar>u' x\<bar>" by auto
2076     also have "\<dots> \<le> w x + w x"
2077       using * by (intro add_mono) auto
2078     finally show "\<bar>u j x - u' x\<bar> \<le> 2 * w x" by simp
2079   qed
2081   have PI_diff: "\<And>n. (\<integral>\<^isup>+ x. ereal (?diff n x) \<partial>M) =
2082     (\<integral>\<^isup>+ x. ereal (2 * w x) \<partial>M) - (\<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
2083     using diff w diff_less_2w w_pos
2084     by (subst positive_integral_diff[symmetric])
2085        (auto simp: integrable_def intro!: positive_integral_cong_AE)
2087   have "integrable M (\<lambda>x. 2 * w x)"
2088     using w by auto
2089   hence I2w_fin: "(\<integral>\<^isup>+ x. ereal (2 * w x) \<partial>M) \<noteq> \<infinity>" and
2090     borel_2w: "(\<lambda>x. ereal (2 * w x)) \<in> borel_measurable M"
2091     unfolding integrable_def by auto
2093   have "limsup (\<lambda>n. \<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M) = 0" (is "limsup ?f = 0")
2094   proof cases
2095     assume eq_0: "(\<integral>\<^isup>+ x. max 0 (ereal (2 * w x)) \<partial>M) = 0" (is "?wx = 0")
2096     { fix n
2097       have "?f n \<le> ?wx" (is "integral\<^isup>P M ?f' \<le> _")
2098         using diff_less_2w unfolding positive_integral_max_0
2099         by (intro positive_integral_mono_AE) auto
2100       then have "?f n = 0"
2101         using positive_integral_positive[of M ?f'] eq_0 by auto }
2102     then show ?thesis by (simp add: Limsup_const)
2103   next
2104     assume neq_0: "(\<integral>\<^isup>+ x. max 0 (ereal (2 * w x)) \<partial>M) \<noteq> 0" (is "?wx \<noteq> 0")
2105     have "0 = limsup (\<lambda>n. 0 :: ereal)" by (simp add: Limsup_const)
2106     also have "\<dots> \<le> limsup (\<lambda>n. \<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
2107       by (intro limsup_mono positive_integral_positive)
2108     finally have pos: "0 \<le> limsup (\<lambda>n. \<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)" .
2109     have "?wx = (\<integral>\<^isup>+ x. liminf (\<lambda>n. max 0 (ereal (?diff n x))) \<partial>M)"
2110       using u'
2111     proof (intro positive_integral_cong_AE, eventually_elim)
2112       fix x assume u': "(\<lambda>i. u i x) ----> u' x"
2113       show "max 0 (ereal (2 * w x)) = liminf (\<lambda>n. max 0 (ereal (?diff n x)))"
2114         unfolding ereal_max_0
2115       proof (rule lim_imp_Liminf[symmetric], unfold lim_ereal)
2116         have "(\<lambda>i. ?diff i x) ----> 2 * w x - \<bar>u' x - u' x\<bar>"
2117           using u' by (safe intro!: tendsto_intros)
2118         then show "(\<lambda>i. max 0 (?diff i x)) ----> max 0 (2 * w x)"
2119           by (auto intro!: tendsto_real_max)
2120       qed (rule trivial_limit_sequentially)
2121     qed
2122     also have "\<dots> \<le> liminf (\<lambda>n. \<integral>\<^isup>+ x. max 0 (ereal (?diff n x)) \<partial>M)"
2123       using borel w u unfolding integrable_def
2124       by (intro positive_integral_lim_INF) (auto intro!: positive_integral_lim_INF)
2125     also have "\<dots> = (\<integral>\<^isup>+ x. ereal (2 * w x) \<partial>M) -
2126         limsup (\<lambda>n. \<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"
2127       unfolding PI_diff positive_integral_max_0
2128       using positive_integral_positive[of M "\<lambda>x. ereal (2 * w x)"]
2129       by (subst liminf_ereal_cminus) auto
2130     finally show ?thesis
2131       using neq_0 I2w_fin positive_integral_positive[of M "\<lambda>x. ereal (2 * w x)"] pos
2132       unfolding positive_integral_max_0
2133       by (cases rule: ereal2_cases[of "\<integral>\<^isup>+ x. ereal (2 * w x) \<partial>M" "limsup (\<lambda>n. \<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M)"])
2134          auto
2135   qed
2137   have "liminf ?f \<le> limsup ?f"
2138     by (intro ereal_Liminf_le_Limsup trivial_limit_sequentially)
2139   moreover
2140   { have "0 = liminf (\<lambda>n. 0 :: ereal)" by (simp add: Liminf_const)
2141     also have "\<dots> \<le> liminf ?f"
2142       by (intro liminf_mono positive_integral_positive)
2143     finally have "0 \<le> liminf ?f" . }
2144   ultimately have liminf_limsup_eq: "liminf ?f = ereal 0" "limsup ?f = ereal 0"
2145     using `limsup ?f = 0` by auto
2146   have "\<And>n. (\<integral>\<^isup>+ x. ereal \<bar>u n x - u' x\<bar> \<partial>M) = ereal (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)"
2147     using diff positive_integral_positive[of M]
2148     by (subst integral_eq_positive_integral[of _ M]) (auto simp: ereal_real integrable_def)
2149   then show ?lim_diff
2150     using ereal_Liminf_eq_Limsup[OF trivial_limit_sequentially liminf_limsup_eq]
2151     by simp
2153   show ?lim
2154   proof (rule LIMSEQ_I)
2155     fix r :: real assume "0 < r"
2156     from LIMSEQ_D[OF `?lim_diff` this]
2157     obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M) < r"
2158       using diff by (auto simp: integral_positive)
2160     show "\<exists>N. \<forall>n\<ge>N. norm (integral\<^isup>L M (u n) - integral\<^isup>L M u') < r"
2161     proof (safe intro!: exI[of _ N])
2162       fix n assume "N \<le> n"
2163       have "\<bar>integral\<^isup>L M (u n) - integral\<^isup>L M u'\<bar> = \<bar>(\<integral>x. u n x - u' x \<partial>M)\<bar>"
2164         using u `integrable M u'` by auto
2165       also have "\<dots> \<le> (\<integral>x. \<bar>u n x - u' x\<bar> \<partial>M)" using u `integrable M u'`
2166         by (rule_tac integral_triangle_inequality) auto
2167       also note N[OF `N \<le> n`]
2168       finally show "norm (integral\<^isup>L M (u n) - integral\<^isup>L M u') < r" by simp
2169     qed
2170   qed
2171 qed
2173 lemma integral_sums:
2174   assumes borel: "\<And>i. integrable M (f i)"
2175   and summable: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>f i x\<bar>)"
2176   and sums: "summable (\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
2177   shows "integrable M (\<lambda>x. (\<Sum>i. f i x))" (is "integrable M ?S")
2178   and "(\<lambda>i. integral\<^isup>L M (f i)) sums (\<integral>x. (\<Sum>i. f i x) \<partial>M)" (is ?integral)
2179 proof -
2180   have "\<forall>x\<in>space M. \<exists>w. (\<lambda>i. \<bar>f i x\<bar>) sums w"
2181     using summable unfolding summable_def by auto
2182   from bchoice[OF this]
2183   obtain w where w: "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. \<bar>f i x\<bar>) sums w x" by auto
2184   then have w_borel: "w \<in> borel_measurable M" unfolding sums_def
2185     by (rule borel_measurable_LIMSEQ) (auto simp: borel[THEN integrableD(1)])
2187   let ?w = "\<lambda>y. if y \<in> space M then w y else 0"
2189   obtain x where abs_sum: "(\<lambda>i. (\<integral>x. \<bar>f i x\<bar> \<partial>M)) sums x"
2190     using sums unfolding summable_def ..
2192   have 1: "\<And>n. integrable M (\<lambda>x. \<Sum>i = 0..<n. f i x)"
2193     using borel by auto
2195   have 2: "\<And>j. AE x in M. \<bar>\<Sum>i = 0..<j. f i x\<bar> \<le> ?w x"
2196     using AE_space
2197   proof eventually_elim
2198     fix j x assume [simp]: "x \<in> space M"
2199     have "\<bar>\<Sum>i = 0..< j. f i x\<bar> \<le> (\<Sum>i = 0..< j. \<bar>f i x\<bar>)" by (rule setsum_abs)
2200     also have "\<dots> \<le> w x" using w[of x] series_pos_le[of "\<lambda>i. \<bar>f i x\<bar>"] unfolding sums_iff by auto
2201     finally show "\<bar>\<Sum>i = 0..<j. f i x\<bar> \<le> ?w x" by simp
2202   qed
2204   have 3: "integrable M ?w"
2205   proof (rule integral_monotone_convergence(1))
2206     let ?F = "\<lambda>n y. (\<Sum>i = 0..<n. \<bar>f i y\<bar>)"
2207     let ?w' = "\<lambda>n y. if y \<in> space M then ?F n y else 0"
2208     have "\<And>n. integrable M (?F n)"
2209       using borel by (auto intro!: integrable_abs)
2210     thus "\<And>n. integrable M (?w' n)" by (simp cong: integrable_cong)
2211     show "AE x in M. mono (\<lambda>n. ?w' n x)"
2212       by (auto simp: mono_def le_fun_def intro!: setsum_mono2)
2213     show "AE x in M. (\<lambda>n. ?w' n x) ----> ?w x"
2214         using w by (simp_all add: tendsto_const sums_def)
2215     have *: "\<And>n. integral\<^isup>L M (?w' n) = (\<Sum>i = 0..< n. (\<integral>x. \<bar>f i x\<bar> \<partial>M))"
2216       using borel by (simp add: integrable_abs cong: integral_cong)
2217     from abs_sum
2218     show "(\<lambda>i. integral\<^isup>L M (?w' i)) ----> x" unfolding * sums_def .
2219   qed (simp add: w_borel measurable_If_set)
2221   from summable[THEN summable_rabs_cancel]
2222   have 4: "AE x in M. (\<lambda>n. \<Sum>i = 0..<n. f i x) ----> (\<Sum>i. f i x)"
2223     by (auto intro: summable_sumr_LIMSEQ_suminf)
2225   note int = integral_dominated_convergence(1,3)[OF 1 2 3 4
2226     borel_measurable_suminf[OF integrableD(1)[OF borel]]]
2228   from int show "integrable M ?S" by simp
2230   show ?integral unfolding sums_def integral_setsum(1)[symmetric, OF borel]
2231     using int(2) by simp
2232 qed
2234 section "Lebesgue integration on countable spaces"
2236 lemma integral_on_countable:
2237   assumes f: "f \<in> borel_measurable M"
2238   and bij: "bij_betw enum S (f ` space M)"
2239   and enum_zero: "enum ` (-S) \<subseteq> {0}"
2240   and fin: "\<And>x. x \<noteq> 0 \<Longrightarrow> (emeasure M) (f -` {x} \<inter> space M) \<noteq> \<infinity>"
2241   and abs_summable: "summable (\<lambda>r. \<bar>enum r * real ((emeasure M) (f -` {enum r} \<inter> space M))\<bar>)"
2242   shows "integrable M f"
2243   and "(\<lambda>r. enum r * real ((emeasure M) (f -` {enum r} \<inter> space M))) sums integral\<^isup>L M f" (is ?sums)
2244 proof -
2245   let ?A = "\<lambda>r. f -` {enum r} \<inter> space M"
2246   let ?F = "\<lambda>r x. enum r * indicator (?A r) x"
2247   have enum_eq: "\<And>r. enum r * real ((emeasure M) (?A r)) = integral\<^isup>L M (?F r)"
2248     using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
2250   { fix x assume "x \<in> space M"
2251     hence "f x \<in> enum ` S" using bij unfolding bij_betw_def by auto
2252     then obtain i where "i\<in>S" "enum i = f x" by auto
2253     have F: "\<And>j. ?F j x = (if j = i then f x else 0)"
2254     proof cases
2255       fix j assume "j = i"
2256       thus "?thesis j" using `x \<in> space M` `enum i = f x` by (simp add: indicator_def)
2257     next
2258       fix j assume "j \<noteq> i"
2259       show "?thesis j" using bij `i \<in> S` `j \<noteq> i` `enum i = f x` enum_zero
2260         by (cases "j \<in> S") (auto simp add: indicator_def bij_betw_def inj_on_def)
2261     qed
2262     hence F_abs: "\<And>j. \<bar>if j = i then f x else 0\<bar> = (if j = i then \<bar>f x\<bar> else 0)" by auto
2263     have "(\<lambda>i. ?F i x) sums f x"
2264          "(\<lambda>i. \<bar>?F i x\<bar>) sums \<bar>f x\<bar>"
2265       by (auto intro!: sums_single simp: F F_abs) }
2266   note F_sums_f = this(1) and F_abs_sums_f = this(2)
2268   have int_f: "integral\<^isup>L M f = (\<integral>x. (\<Sum>r. ?F r x) \<partial>M)" "integrable M f = integrable M (\<lambda>x. \<Sum>r. ?F r x)"
2269     using F_sums_f by (auto intro!: integral_cong integrable_cong simp: sums_iff)
2271   { fix r
2272     have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = (\<integral>x. \<bar>enum r\<bar> * indicator (?A r) x \<partial>M)"
2273       by (auto simp: indicator_def intro!: integral_cong)
2274     also have "\<dots> = \<bar>enum r\<bar> * real ((emeasure M) (?A r))"
2275       using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
2276     finally have "(\<integral>x. \<bar>?F r x\<bar> \<partial>M) = \<bar>enum r * real ((emeasure M) (?A r))\<bar>"
2277       using f by (subst (2) abs_mult_pos[symmetric]) (auto intro!: real_of_ereal_pos measurable_sets) }
2278   note int_abs_F = this
2280   have 1: "\<And>i. integrable M (\<lambda>x. ?F i x)"
2281     using f fin by (simp add: borel_measurable_vimage integral_cmul_indicator)
2283   have 2: "\<And>x. x \<in> space M \<Longrightarrow> summable (\<lambda>i. \<bar>?F i x\<bar>)"
2284     using F_abs_sums_f unfolding sums_iff by auto
2286   from integral_sums(2)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
2287   show ?sums unfolding enum_eq int_f by simp
2289   from integral_sums(1)[OF 1 2, unfolded int_abs_F, OF _ abs_summable]
2290   show "integrable M f" unfolding int_f by simp
2291 qed
2293 section {* Distributions *}
2295 lemma positive_integral_distr':
2296   assumes T: "T \<in> measurable M M'"
2297   and f: "f \<in> borel_measurable (distr M M' T)" "\<And>x. 0 \<le> f x"
2298   shows "integral\<^isup>P (distr M M' T) f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)"
2299   using f
2300 proof induct
2301   case (cong f g)
2302   with T show ?case
2303     apply (subst positive_integral_cong[of _ f g])
2304     apply simp
2305     apply (subst positive_integral_cong[of _ "\<lambda>x. f (T x)" "\<lambda>x. g (T x)"])
2306     apply (simp add: measurable_def Pi_iff)
2307     apply simp
2308     done
2309 next
2310   case (set A)
2311   then have eq: "\<And>x. x \<in> space M \<Longrightarrow> indicator A (T x) = indicator (T -` A \<inter> space M) x"
2312     by (auto simp: indicator_def)
2313   from set T show ?case
2314     by (subst positive_integral_cong[OF eq])
2315        (auto simp add: emeasure_distr intro!: positive_integral_indicator[symmetric] measurable_sets)
2316 qed (simp_all add: measurable_compose[OF T] T positive_integral_cmult positive_integral_add
2317                    positive_integral_monotone_convergence_SUP le_fun_def incseq_def)
2319 lemma positive_integral_distr:
2320   assumes T: "T \<in> measurable M M'"
2321   and f: "f \<in> borel_measurable M'"
2322   shows "integral\<^isup>P (distr M M' T) f = (\<integral>\<^isup>+ x. f (T x) \<partial>M)"
2323   apply (subst (1 2) positive_integral_max_0[symmetric])
2324   apply (rule positive_integral_distr')
2325   apply (auto simp: f T)
2326   done
2328 lemma integral_distr:
2329   "T \<in> measurable M M' \<Longrightarrow> f \<in> borel_measurable M' \<Longrightarrow> integral\<^isup>L (distr M M' T) f = (\<integral> x. f (T x) \<partial>M)"
2330   unfolding lebesgue_integral_def
2331   by (subst (1 2) positive_integral_distr) auto
2333 lemma integrable_distr_eq:
2334   assumes T: "T \<in> measurable M M'" "f \<in> borel_measurable M'"
2335   shows "integrable (distr M M' T) f \<longleftrightarrow> integrable M (\<lambda>x. f (T x))"
2336   using T measurable_comp[OF T]
2337   unfolding integrable_def
2338   by (subst (1 2) positive_integral_distr) (auto simp: comp_def)
2340 lemma integrable_distr:
2341   assumes T: "T \<in> measurable M M'" shows "integrable (distr M M' T) f \<Longrightarrow> integrable M (\<lambda>x. f (T x))"
2342   by (subst integrable_distr_eq[symmetric, OF T]) auto
2344 section {* Lebesgue integration on @{const count_space} *}
2346 lemma simple_function_count_space[simp]:
2347   "simple_function (count_space A) f \<longleftrightarrow> finite (f ` A)"
2348   unfolding simple_function_def by simp
2350 lemma positive_integral_count_space:
2351   assumes A: "finite {a\<in>A. 0 < f a}"
2352   shows "integral\<^isup>P (count_space A) f = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)"
2353 proof -
2354   have *: "(\<integral>\<^isup>+x. max 0 (f x) \<partial>count_space A) =
2355     (\<integral>\<^isup>+ x. (\<Sum>a|a\<in>A \<and> 0 < f a. f a * indicator {a} x) \<partial>count_space A)"
2356     by (auto intro!: positive_integral_cong
2357              simp add: indicator_def if_distrib setsum_cases[OF A] max_def le_less)
2358   also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. \<integral>\<^isup>+ x. f a * indicator {a} x \<partial>count_space A)"
2359     by (subst positive_integral_setsum)
2360        (simp_all add: AE_count_space ereal_zero_le_0_iff less_imp_le)
2361   also have "\<dots> = (\<Sum>a|a\<in>A \<and> 0 < f a. f a)"
2362     by (auto intro!: setsum_cong simp: positive_integral_cmult_indicator one_ereal_def[symmetric])
2363   finally show ?thesis by (simp add: positive_integral_max_0)
2364 qed
2366 lemma integrable_count_space:
2367   "finite X \<Longrightarrow> integrable (count_space X) f"
2368   by (auto simp: positive_integral_count_space integrable_def)
2370 lemma positive_integral_count_space_finite:
2371     "finite A \<Longrightarrow> (\<integral>\<^isup>+x. f x \<partial>count_space A) = (\<Sum>a\<in>A. max 0 (f a))"
2372   by (subst positive_integral_max_0[symmetric])
2373      (auto intro!: setsum_mono_zero_left simp: positive_integral_count_space less_le)
2375 lemma lebesgue_integral_count_space_finite_support:
2376   assumes f: "finite {a\<in>A. f a \<noteq> 0}" shows "(\<integral>x. f x \<partial>count_space A) = (\<Sum>a | a \<in> A \<and> f a \<noteq> 0. f a)"
2377 proof -
2378   have *: "\<And>r::real. 0 < max 0 r \<longleftrightarrow> 0 < r" "\<And>x. max 0 (ereal x) = ereal (max 0 x)"
2379     "\<And>a. a \<in> A \<and> 0 < f a \<Longrightarrow> max 0 (f a) = f a"
2380     "\<And>a. a \<in> A \<and> f a < 0 \<Longrightarrow> max 0 (- f a) = - f a"
2381     "{a \<in> A. f a \<noteq> 0} = {a \<in> A. 0 < f a} \<union> {a \<in> A. f a < 0}"
2382     "({a \<in> A. 0 < f a} \<inter> {a \<in> A. f a < 0}) = {}"
2383     by (auto split: split_max)
2384   have "finite {a \<in> A. 0 < f a}" "finite {a \<in> A. f a < 0}"
2385     by (auto intro: finite_subset[OF _ f])
2386   then show ?thesis
2387     unfolding lebesgue_integral_def
2388     apply (subst (1 2) positive_integral_max_0[symmetric])
2389     apply (subst (1 2) positive_integral_count_space)
2390     apply (auto simp add: * setsum_negf setsum_Un
2391                 simp del: ereal_max)
2392     done
2393 qed
2395 lemma lebesgue_integral_count_space_finite:
2396     "finite A \<Longrightarrow> (\<integral>x. f x \<partial>count_space A) = (\<Sum>a\<in>A. f a)"
2397   apply (auto intro!: setsum_mono_zero_left
2398               simp: positive_integral_count_space_finite lebesgue_integral_def)
2399   apply (subst (1 2)  setsum_real_of_ereal[symmetric])
2400   apply (auto simp: max_def setsum_subtractf[symmetric] intro!: setsum_cong)
2401   done
2403 lemma borel_measurable_count_space[simp, intro!]:
2404   "f \<in> borel_measurable (count_space A)"
2405   by simp
2407 section {* Measure spaces with an associated density *}
2409 definition density :: "'a measure \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
2410   "density M f = measure_of (space M) (sets M) (\<lambda>A. \<integral>\<^isup>+ x. f x * indicator A x \<partial>M)"
2412 lemma
2413   shows sets_density[simp]: "sets (density M f) = sets M"
2414     and space_density[simp]: "space (density M f) = space M"
2415   by (auto simp: density_def)
2417 lemma
2418   shows measurable_density_eq1[simp]: "g \<in> measurable (density Mg f) Mg' \<longleftrightarrow> g \<in> measurable Mg Mg'"
2419     and measurable_density_eq2[simp]: "h \<in> measurable Mh (density Mh' f) \<longleftrightarrow> h \<in> measurable Mh Mh'"
2420     and simple_function_density_eq[simp]: "simple_function (density Mu f) u \<longleftrightarrow> simple_function Mu u"
2421   unfolding measurable_def simple_function_def by simp_all
2423 lemma density_cong: "f \<in> borel_measurable M \<Longrightarrow> f' \<in> borel_measurable M \<Longrightarrow>
2424   (AE x in M. f x = f' x) \<Longrightarrow> density M f = density M f'"
2425   unfolding density_def by (auto intro!: measure_of_eq positive_integral_cong_AE space_closed)
2427 lemma density_max_0: "density M f = density M (\<lambda>x. max 0 (f x))"
2428 proof -
2429   have "\<And>x A. max 0 (f x) * indicator A x = max 0 (f x * indicator A x)"
2430     by (auto simp: indicator_def)
2431   then show ?thesis
2432     unfolding density_def by (simp add: positive_integral_max_0)
2433 qed
2435 lemma density_ereal_max_0: "density M (\<lambda>x. ereal (f x)) = density M (\<lambda>x. ereal (max 0 (f x)))"
2436   by (subst density_max_0) (auto intro!: arg_cong[where f="density M"] split: split_max)
2438 lemma emeasure_density:
2439   assumes f[measurable]: "f \<in> borel_measurable M" and A[measurable]: "A \<in> sets M"
2440   shows "emeasure (density M f) A = (\<integral>\<^isup>+ x. f x * indicator A x \<partial>M)"
2441     (is "_ = ?\<mu> A")
2442   unfolding density_def
2443 proof (rule emeasure_measure_of_sigma)
2444   show "sigma_algebra (space M) (sets M)" ..
2445   show "positive (sets M) ?\<mu>"
2446     using f by (auto simp: positive_def intro!: positive_integral_positive)
2447   have \<mu>_eq: "?\<mu> = (\<lambda>A. \<integral>\<^isup>+ x. max 0 (f x) * indicator A x \<partial>M)" (is "?\<mu> = ?\<mu>'")
2448     apply (subst positive_integral_max_0[symmetric])
2449     apply (intro ext positive_integral_cong_AE AE_I2)
2450     apply (auto simp: indicator_def)
2451     done
2452   show "countably_additive (sets M) ?\<mu>"
2453     unfolding \<mu>_eq
2454   proof (intro countably_additiveI)
2455     fix A :: "nat \<Rightarrow> 'a set" assume "range A \<subseteq> sets M"
2456     then have "\<And>i. A i \<in> sets M" by auto
2457     then have *: "\<And>i. (\<lambda>x. max 0 (f x) * indicator (A i) x) \<in> borel_measurable M"
2458       by (auto simp: set_eq_iff)
2459     assume disj: "disjoint_family A"
2460     have "(\<Sum>n. ?\<mu>' (A n)) = (\<integral>\<^isup>+ x. (\<Sum>n. max 0 (f x) * indicator (A n) x) \<partial>M)"
2461       using f * by (simp add: positive_integral_suminf)
2462     also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) * (\<Sum>n. indicator (A n) x) \<partial>M)" using f
2463       by (auto intro!: suminf_cmult_ereal positive_integral_cong_AE)
2464     also have "\<dots> = (\<integral>\<^isup>+ x. max 0 (f x) * indicator (\<Union>n. A n) x \<partial>M)"
2465       unfolding suminf_indicator[OF disj] ..
2466     finally show "(\<Sum>n. ?\<mu>' (A n)) = ?\<mu>' (\<Union>x. A x)" by simp
2467   qed
2468 qed fact
2470 lemma null_sets_density_iff:
2471   assumes f: "f \<in> borel_measurable M"
2472   shows "A \<in> null_sets (density M f) \<longleftrightarrow> A \<in> sets M \<and> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)"
2473 proof -
2474   { assume "A \<in> sets M"
2475     have eq: "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = (\<integral>\<^isup>+x. max 0 (f x) * indicator A x \<partial>M)"
2476       apply (subst positive_integral_max_0[symmetric])
2477       apply (intro positive_integral_cong)
2478       apply (auto simp: indicator_def)
2479       done
2480     have "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow>
2481       emeasure M {x \<in> space M. max 0 (f x) * indicator A x \<noteq> 0} = 0"
2482       unfolding eq
2483       using f `A \<in> sets M`
2484       by (intro positive_integral_0_iff) auto
2485     also have "\<dots> \<longleftrightarrow> (AE x in M. max 0 (f x) * indicator A x = 0)"
2486       using f `A \<in> sets M`
2487       by (intro AE_iff_measurable[OF _ refl, symmetric]) auto
2488     also have "(AE x in M. max 0 (f x) * indicator A x = 0) \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)"
2489       by (auto simp add: indicator_def max_def split: split_if_asm)
2490     finally have "(\<integral>\<^isup>+x. f x * indicator A x \<partial>M) = 0 \<longleftrightarrow> (AE x in M. x \<in> A \<longrightarrow> f x \<le> 0)" . }
2491   with f show ?thesis
2492     by (simp add: null_sets_def emeasure_density cong: conj_cong)
2493 qed
2495 lemma AE_density:
2496   assumes f: "f \<in> borel_measurable M"
2497   shows "(AE x in density M f. P x) \<longleftrightarrow> (AE x in M. 0 < f x \<longrightarrow> P x)"
2498 proof
2499   assume "AE x in density M f. P x"
2500   with f obtain N where "{x \<in> space M. \<not> P x} \<subseteq> N" "N \<in> sets M" and ae: "AE x in M. x \<in> N \<longrightarrow> f x \<le> 0"
2501     by (auto simp: eventually_ae_filter null_sets_density_iff)
2502   then have "AE x in M. x \<notin> N \<longrightarrow> P x" by auto
2503   with ae show "AE x in M. 0 < f x \<longrightarrow> P x"
2504     by (rule eventually_elim2) auto
2505 next
2506   fix N assume ae: "AE x in M. 0 < f x \<longrightarrow> P x"
2507   then obtain N where "{x \<in> space M. \<not> (0 < f x \<longrightarrow> P x)} \<subseteq> N" "N \<in> null_sets M"
2508     by (auto simp: eventually_ae_filter)
2509   then have *: "{x \<in> space (density M f). \<not> P x} \<subseteq> N \<union> {x\<in>space M. \<not> 0 < f x}"
2510     "N \<union> {x\<in>space M. \<not> 0 < f x} \<in> sets M" and ae2: "AE x in M. x \<notin> N"
2511     using f by (auto simp: subset_eq intro!: sets_Collect_neg AE_not_in)
2512   show "AE x in density M f. P x"
2513     using ae2
2514     unfolding eventually_ae_filter[of _ "density M f"] Bex_def null_sets_density_iff[OF f]
2515     by (intro exI[of _ "N \<union> {x\<in>space M. \<not> 0 < f x}"] conjI *)
2516        (auto elim: eventually_elim2)
2517 qed
2519 lemma positive_integral_density':
2520   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
2521   assumes g: "g \<in> borel_measurable M" "\<And>x. 0 \<le> g x"
2522   shows "integral\<^isup>P (density M f) g = (\<integral>\<^isup>+ x. f x * g x \<partial>M)"
2523 using g proof induct
2524   case (cong u v)
2525   then show ?case
2526     apply (subst positive_integral_cong[OF cong(3)])
2527     apply (simp_all cong: positive_integral_cong)
2528     done
2529 next
2530   case (set A) then show ?case
2531     by (simp add: emeasure_density f)
2532 next
2533   case (mult u c)
2534   moreover have "\<And>x. f x * (c * u x) = c * (f x * u x)" by (simp add: field_simps)
2535   ultimately show ?case
2536     by (simp add: f positive_integral_cmult)
2537 next
2538   case (add u v)
2539   moreover then have "\<And>x. f x * (v x + u x) = f x * v x + f x * u x"
2540     by (simp add: ereal_right_distrib)
2541   moreover note f
2542   ultimately show ?case
2544 next
2545   case (seq U)
2546   from f(2) have eq: "AE x in M. f x * (SUP i. U i x) = (SUP i. f x * U i x)"
2547     by eventually_elim (simp add: SUPR_ereal_cmult seq)
2548   from seq f show ?case
2549     apply (simp add: positive_integral_monotone_convergence_SUP)
2550     apply (subst positive_integral_cong_AE[OF eq])
2551     apply (subst positive_integral_monotone_convergence_SUP_AE)
2552     apply (auto simp: incseq_def le_fun_def intro!: ereal_mult_left_mono)
2553     done
2554 qed
2556 lemma positive_integral_density:
2557   "f \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow> g' \<in> borel_measurable M \<Longrightarrow>
2558     integral\<^isup>P (density M f) g' = (\<integral>\<^isup>+ x. f x * g' x \<partial>M)"
2559   by (subst (1 2) positive_integral_max_0[symmetric])
2560      (auto intro!: positive_integral_cong_AE
2561            simp: measurable_If max_def ereal_zero_le_0_iff positive_integral_density')
2563 lemma integral_density:
2564   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
2565     and g: "g \<in> borel_measurable M"
2566   shows "integral\<^isup>L (density M f) g = (\<integral> x. f x * g x \<partial>M)"
2567     and "integrable (density M f) g \<longleftrightarrow> integrable M (\<lambda>x. f x * g x)"
2568   unfolding lebesgue_integral_def integrable_def using f g
2569   by (auto simp: positive_integral_density)
2571 lemma emeasure_restricted:
2572   assumes S: "S \<in> sets M" and X: "X \<in> sets M"
2573   shows "emeasure (density M (indicator S)) X = emeasure M (S \<inter> X)"
2574 proof -
2575   have "emeasure (density M (indicator S)) X = (\<integral>\<^isup>+x. indicator S x * indicator X x \<partial>M)"
2576     using S X by (simp add: emeasure_density)
2577   also have "\<dots> = (\<integral>\<^isup>+x. indicator (S \<inter> X) x \<partial>M)"
2578     by (auto intro!: positive_integral_cong simp: indicator_def)
2579   also have "\<dots> = emeasure M (S \<inter> X)"
2580     using S X by (simp add: Int)
2581   finally show ?thesis .
2582 qed
2584 lemma measure_restricted:
2585   "S \<in> sets M \<Longrightarrow> X \<in> sets M \<Longrightarrow> measure (density M (indicator S)) X = measure M (S \<inter> X)"
2586   by (simp add: emeasure_restricted measure_def)
2588 lemma (in finite_measure) finite_measure_restricted:
2589   "S \<in> sets M \<Longrightarrow> finite_measure (density M (indicator S))"
2590   by default (simp add: emeasure_restricted)
2592 lemma emeasure_density_const:
2593   "A \<in> sets M \<Longrightarrow> 0 \<le> c \<Longrightarrow> emeasure (density M (\<lambda>_. c)) A = c * emeasure M A"
2594   by (auto simp: positive_integral_cmult_indicator emeasure_density)
2596 lemma measure_density_const:
2597   "A \<in> sets M \<Longrightarrow> 0 < c \<Longrightarrow> c \<noteq> \<infinity> \<Longrightarrow> measure (density M (\<lambda>_. c)) A = real c * measure M A"
2598   by (auto simp: emeasure_density_const measure_def)
2600 lemma density_density_eq:
2601    "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow> AE x in M. 0 \<le> f x \<Longrightarrow>
2602    density (density M f) g = density M (\<lambda>x. f x * g x)"
2603   by (auto intro!: measure_eqI simp: emeasure_density positive_integral_density ac_simps)
2605 lemma distr_density_distr:
2606   assumes T: "T \<in> measurable M M'" and T': "T' \<in> measurable M' M"
2607     and inv: "\<forall>x\<in>space M. T' (T x) = x"
2608   assumes f: "f \<in> borel_measurable M'"
2609   shows "distr (density (distr M M' T) f) M T' = density M (f \<circ> T)" (is "?R = ?L")
2610 proof (rule measure_eqI)
2611   fix A assume A: "A \<in> sets ?R"
2612   { fix x assume "x \<in> space M"
2613     with sets_into_space[OF A]
2614     have "indicator (T' -` A \<inter> space M') (T x) = (indicator A x :: ereal)"
2615       using T inv by (auto simp: indicator_def measurable_space) }
2616   with A T T' f show "emeasure ?R A = emeasure ?L A"
2617     by (simp add: measurable_comp emeasure_density emeasure_distr
2618                   positive_integral_distr measurable_sets cong: positive_integral_cong)
2619 qed simp
2621 lemma density_density_divide:
2622   fixes f g :: "'a \<Rightarrow> real"
2623   assumes f: "f \<in> borel_measurable M" "AE x in M. 0 \<le> f x"
2624   assumes g: "g \<in> borel_measurable M" "AE x in M. 0 \<le> g x"
2625   assumes ac: "AE x in M. f x = 0 \<longrightarrow> g x = 0"
2626   shows "density (density M f) (\<lambda>x. g x / f x) = density M g"
2627 proof -
2628   have "density M g = density M (\<lambda>x. f x * (g x / f x))"
2629     using f g ac by (auto intro!: density_cong measurable_If)
2630   then show ?thesis
2631     using f g by (subst density_density_eq) auto
2632 qed
2634 section {* Point measure *}
2636 definition point_measure :: "'a set \<Rightarrow> ('a \<Rightarrow> ereal) \<Rightarrow> 'a measure" where
2637   "point_measure A f = density (count_space A) f"
2639 lemma
2640   shows space_point_measure: "space (point_measure A f) = A"
2641     and sets_point_measure: "sets (point_measure A f) = Pow A"
2642   by (auto simp: point_measure_def)
2644 lemma measurable_point_measure_eq1[simp]:
2645   "g \<in> measurable (point_measure A f) M \<longleftrightarrow> g \<in> A \<rightarrow> space M"
2646   unfolding point_measure_def by simp
2648 lemma measurable_point_measure_eq2_finite[simp]:
2649   "finite A \<Longrightarrow>
2650    g \<in> measurable M (point_measure A f) \<longleftrightarrow>
2651     (g \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. g -` {a} \<inter> space M \<in> sets M))"
2652   unfolding point_measure_def by (simp add: measurable_count_space_eq2)
2654 lemma simple_function_point_measure[simp]:
2655   "simple_function (point_measure A f) g \<longleftrightarrow> finite (g ` A)"
2656   by (simp add: point_measure_def)
2658 lemma emeasure_point_measure:
2659   assumes A: "finite {a\<in>X. 0 < f a}" "X \<subseteq> A"
2660   shows "emeasure (point_measure A f) X = (\<Sum>a|a\<in>X \<and> 0 < f a. f a)"
2661 proof -
2662   have "{a. (a \<in> X \<longrightarrow> a \<in> A \<and> 0 < f a) \<and> a \<in> X} = {a\<in>X. 0 < f a}"
2663     using `X \<subseteq> A` by auto
2664   with A show ?thesis
2665     by (simp add: emeasure_density positive_integral_count_space ereal_zero_le_0_iff
2666                   point_measure_def indicator_def)
2667 qed
2669 lemma emeasure_point_measure_finite:
2670   "finite A \<Longrightarrow> (\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (point_measure A f) X = (\<Sum>a\<in>X. f a)"
2671   by (subst emeasure_point_measure) (auto dest: finite_subset intro!: setsum_mono_zero_left simp: less_le)
2673 lemma emeasure_point_measure_finite2:
2674   "X \<subseteq> A \<Longrightarrow> finite X \<Longrightarrow> (\<And>i. i \<in> X \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> emeasure (point_measure A f) X = (\<Sum>a\<in>X. f a)"
2675   by (subst emeasure_point_measure)
2676      (auto dest: finite_subset intro!: setsum_mono_zero_left simp: less_le)
2678 lemma null_sets_point_measure_iff:
2679   "X \<in> null_sets (point_measure A f) \<longleftrightarrow> X \<subseteq> A \<and> (\<forall>x\<in>X. f x \<le> 0)"
2680  by (auto simp: AE_count_space null_sets_density_iff point_measure_def)
2682 lemma AE_point_measure:
2683   "(AE x in point_measure A f. P x) \<longleftrightarrow> (\<forall>x\<in>A. 0 < f x \<longrightarrow> P x)"
2684   unfolding point_measure_def
2685   by (subst AE_density) (auto simp: AE_density AE_count_space point_measure_def)
2687 lemma positive_integral_point_measure:
2688   "finite {a\<in>A. 0 < f a \<and> 0 < g a} \<Longrightarrow>
2689     integral\<^isup>P (point_measure A f) g = (\<Sum>a|a\<in>A \<and> 0 < f a \<and> 0 < g a. f a * g a)"
2690   unfolding point_measure_def
2691   apply (subst density_max_0)
2692   apply (subst positive_integral_density)
2693   apply (simp_all add: AE_count_space positive_integral_density)
2694   apply (subst positive_integral_count_space )
2695   apply (auto intro!: setsum_cong simp: max_def ereal_zero_less_0_iff)
2696   apply (rule finite_subset)
2697   prefer 2
2698   apply assumption
2699   apply auto
2700   done
2702 lemma positive_integral_point_measure_finite:
2703   "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> f a) \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> g a) \<Longrightarrow>
2704     integral\<^isup>P (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)"
2705   by (subst positive_integral_point_measure) (auto intro!: setsum_mono_zero_left simp: less_le)
2707 lemma lebesgue_integral_point_measure_finite:
2708   "finite A \<Longrightarrow> (\<And>a. a \<in> A \<Longrightarrow> 0 \<le> f a) \<Longrightarrow> integral\<^isup>L (point_measure A f) g = (\<Sum>a\<in>A. f a * g a)"
2709   by (simp add: lebesgue_integral_count_space_finite AE_count_space integral_density point_measure_def)
2711 lemma integrable_point_measure_finite:
2712   "finite A \<Longrightarrow> integrable (point_measure A (\<lambda>x. ereal (f x))) g"
2713   unfolding point_measure_def
2714   apply (subst density_ereal_max_0)
2715   apply (subst integral_density)
2716   apply (auto simp: AE_count_space integrable_count_space)
2717   done
2719 section {* Uniform measure *}
2721 definition "uniform_measure M A = density M (\<lambda>x. indicator A x / emeasure M A)"
2723 lemma
2724   shows sets_uniform_measure[simp]: "sets (uniform_measure M A) = sets M"
2725     and space_uniform_measure[simp]: "space (uniform_measure M A) = space M"
2726   by (auto simp: uniform_measure_def)
2728 lemma emeasure_uniform_measure[simp]:
2729   assumes A: "A \<in> sets M" and B: "B \<in> sets M"
2730   shows "emeasure (uniform_measure M A) B = emeasure M (A \<inter> B) / emeasure M A"
2731 proof -
2732   from A B have "emeasure (uniform_measure M A) B = (\<integral>\<^isup>+x. (1 / emeasure M A) * indicator (A \<inter> B) x \<partial>M)"
2733     by (auto simp add: uniform_measure_def emeasure_density split: split_indicator
2734              intro!: positive_integral_cong)
2735   also have "\<dots> = emeasure M (A \<inter> B) / emeasure M A"
2736     using A B
2737     by (subst positive_integral_cmult_indicator) (simp_all add: Int emeasure_nonneg)
2738   finally show ?thesis .
2739 qed
2741 lemma emeasure_neq_0_sets: "emeasure M A \<noteq> 0 \<Longrightarrow> A \<in> sets M"
2742   using emeasure_notin_sets[of A M] by blast
2744 lemma measure_uniform_measure[simp]:
2745   assumes A: "emeasure M A \<noteq> 0" "emeasure M A \<noteq> \<infinity>" and B: "B \<in> sets M"
2746   shows "measure (uniform_measure M A) B = measure M (A \<inter> B) / measure M A"
2747   using emeasure_uniform_measure[OF emeasure_neq_0_sets[OF A(1)] B] A
2748   by (cases "emeasure M A" "emeasure M (A \<inter> B)" rule: ereal2_cases) (simp_all add: measure_def)
2750 section {* Uniform count measure *}
2752 definition "uniform_count_measure A = point_measure A (\<lambda>x. 1 / card A)"
2754 lemma
2755   shows space_uniform_count_measure: "space (uniform_count_measure A) = A"
2756     and sets_uniform_count_measure: "sets (uniform_count_measure A) = Pow A"
2757     unfolding uniform_count_measure_def by (auto simp: space_point_measure sets_point_measure)
2759 lemma emeasure_uniform_count_measure:
2760   "finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> emeasure (uniform_count_measure A) X = card X / card A"
2761   by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def)
2763 lemma measure_uniform_count_measure:
2764   "finite A \<Longrightarrow> X \<subseteq> A \<Longrightarrow> measure (uniform_count_measure A) X = card X / card A"
2765   by (simp add: real_eq_of_nat emeasure_point_measure_finite uniform_count_measure_def measure_def)
2767 end