header {*Lebesgue Integration*}
theory Lebesgue
imports Measure Borel
begin
text{*From the HOL4 Hurd/Coble Lebesgue integration, translated by Armin Heller and Johannes Hoelzl.*}
definition
"pos_part f = (\<lambda>x. max 0 (f x))"
definition
"neg_part f = (\<lambda>x. - min 0 (f x))"
definition
"nonneg f = (\<forall>x. 0 \<le> f x)"
lemma nonneg_pos_part[intro!]:
fixes f :: "'c \<Rightarrow> 'd\<Colon>{linorder,zero}"
shows "nonneg (pos_part f)"
unfolding nonneg_def pos_part_def by auto
lemma nonneg_neg_part[intro!]:
fixes f :: "'c \<Rightarrow> 'd\<Colon>{linorder,ordered_ab_group_add}"
shows "nonneg (neg_part f)"
unfolding nonneg_def neg_part_def min_def by auto
lemma pos_neg_part_abs:
fixes f :: "'a \<Rightarrow> real"
shows "pos_part f x + neg_part f x = \<bar>f x\<bar>"
unfolding real_abs_def pos_part_def neg_part_def by auto
lemma pos_part_abs:
fixes f :: "'a \<Rightarrow> real"
shows "pos_part (\<lambda> x. \<bar>f x\<bar>) y = \<bar>f y\<bar>"
unfolding pos_part_def real_abs_def by auto
lemma neg_part_abs:
fixes f :: "'a \<Rightarrow> real"
shows "neg_part (\<lambda> x. \<bar>f x\<bar>) y = 0"
unfolding neg_part_def real_abs_def by auto
lemma (in measure_space)
assumes "f \<in> borel_measurable M"
shows pos_part_borel_measurable: "pos_part f \<in> borel_measurable M"
and neg_part_borel_measurable: "neg_part f \<in> borel_measurable M"
using assms
proof -
{ fix a :: real
{ assume asm: "0 \<le> a"
from asm have pp: "\<And> w. (pos_part f w \<le> a) = (f w \<le> a)" unfolding pos_part_def by auto
have "{w | w. w \<in> space M \<and> f w \<le> a} \<in> sets M"
unfolding pos_part_def using assms borel_measurable_le_iff by auto
hence "{w . w \<in> space M \<and> pos_part f w \<le> a} \<in> sets M" using pp by auto }
moreover have "a < 0 \<Longrightarrow> {w \<in> space M. pos_part f w \<le> a} \<in> sets M"
unfolding pos_part_def using empty_sets by auto
ultimately have "{w . w \<in> space M \<and> pos_part f w \<le> a} \<in> sets M"
using le_less_linear by auto
} hence pos: "pos_part f \<in> borel_measurable M" using borel_measurable_le_iff by auto
{ fix a :: real
{ assume asm: "0 \<le> a"
from asm have pp: "\<And> w. (neg_part f w \<le> a) = (f w \<ge> - a)" unfolding neg_part_def by auto
have "{w | w. w \<in> space M \<and> f w \<ge> - a} \<in> sets M"
unfolding neg_part_def using assms borel_measurable_ge_iff by auto
hence "{w . w \<in> space M \<and> neg_part f w \<le> a} \<in> sets M" using pp by auto }
moreover have "a < 0 \<Longrightarrow> {w \<in> space M. neg_part f w \<le> a} = {}" unfolding neg_part_def by auto
moreover hence "a < 0 \<Longrightarrow> {w \<in> space M. neg_part f w \<le> a} \<in> sets M" by (simp only: empty_sets)
ultimately have "{w . w \<in> space M \<and> neg_part f w \<le> a} \<in> sets M"
using le_less_linear by auto
} hence neg: "neg_part f \<in> borel_measurable M" using borel_measurable_le_iff by auto
from pos neg show "pos_part f \<in> borel_measurable M" and "neg_part f \<in> borel_measurable M" by auto
qed
lemma (in measure_space) pos_part_neg_part_borel_measurable_iff:
"f \<in> borel_measurable M \<longleftrightarrow>
pos_part f \<in> borel_measurable M \<and> neg_part f \<in> borel_measurable M"
proof -
{ fix x
have "f x = pos_part f x - neg_part f x"
unfolding pos_part_def neg_part_def unfolding max_def min_def
by auto }
hence "(\<lambda> x. f x) = (\<lambda> x. pos_part f x - neg_part f x)" by auto
hence "f = (\<lambda> x. pos_part f x - neg_part f x)" by blast
from pos_part_borel_measurable[of f] neg_part_borel_measurable[of f]
borel_measurable_diff_borel_measurable[of "pos_part f" "neg_part f"]
this
show ?thesis by auto
qed
context measure_space
begin
section "Simple discrete step function"
definition
"pos_simple f =
{ (s :: nat set, a, x).
finite s \<and> nonneg f \<and> nonneg x \<and> a ` s \<subseteq> sets M \<and>
(\<forall>t \<in> space M. (\<exists>!i\<in>s. t\<in>a i) \<and> (\<forall>i\<in>s. t \<in> a i \<longrightarrow> f t = x i)) }"
definition
"pos_simple_integral \<equiv> \<lambda>(s, a, x). \<Sum> i \<in> s. x i * measure M (a i)"
definition
"psfis f = pos_simple_integral ` (pos_simple f)"
lemma pos_simpleE[consumes 1]:
assumes ps: "(s, a, x) \<in> pos_simple f"
obtains "finite s" and "nonneg f" and "nonneg x"
and "a ` s \<subseteq> sets M" and "(\<Union>i\<in>s. a i) = space M"
and "disjoint_family_on a s"
and "\<And>t. t \<in> space M \<Longrightarrow> (\<exists>!i. i \<in> s \<and> t \<in> a i)"
and "\<And>t i. \<lbrakk> t \<in> space M ; i \<in> s ; t \<in> a i \<rbrakk> \<Longrightarrow> f t = x i"
proof
show "finite s" and "nonneg f" and "nonneg x"
and as_in_M: "a ` s \<subseteq> sets M"
and *: "\<And>t. t \<in> space M \<Longrightarrow> (\<exists>!i. i \<in> s \<and> t \<in> a i)"
and **: "\<And>t i. \<lbrakk> t \<in> space M ; i \<in> s ; t \<in> a i \<rbrakk> \<Longrightarrow> f t = x i"
using ps unfolding pos_simple_def by auto
thus t: "(\<Union>i\<in>s. a i) = space M"
proof safe
fix x assume "x \<in> space M"
from *[OF this] show "x \<in> (\<Union>i\<in>s. a i)" by auto
next
fix t i assume "i \<in> s"
hence "a i \<in> sets M" using as_in_M by auto
moreover assume "t \<in> a i"
ultimately show "t \<in> space M" using sets_into_space by blast
qed
show "disjoint_family_on a s" unfolding disjoint_family_on_def
proof safe
fix i j and t assume "i \<in> s" "t \<in> a i" and "j \<in> s" "t \<in> a j" and "i \<noteq> j"
with t * show "t \<in> {}" by auto
qed
qed
lemma pos_simple_cong:
assumes "nonneg f" and "nonneg g" and "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
shows "pos_simple f = pos_simple g"
unfolding pos_simple_def using assms by auto
lemma psfis_cong:
assumes "nonneg f" and "nonneg g" and "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
shows "psfis f = psfis g"
unfolding psfis_def using pos_simple_cong[OF assms] by simp
lemma psfis_0: "0 \<in> psfis (\<lambda>x. 0)"
unfolding psfis_def pos_simple_def pos_simple_integral_def
by (auto simp: nonneg_def
intro: image_eqI[where x="({0}, (\<lambda>i. space M), (\<lambda>i. 0))"])
lemma pos_simple_setsum_indicator_fn:
assumes ps: "(s, a, x) \<in> pos_simple f"
and "t \<in> space M"
shows "(\<Sum>i\<in>s. x i * indicator_fn (a i) t) = f t"
proof -
from assms obtain i where *: "i \<in> s" "t \<in> a i"
and "finite s" and xi: "x i = f t" by (auto elim!: pos_simpleE)
have **: "(\<Sum>i\<in>s. x i * indicator_fn (a i) t) =
(\<Sum>j\<in>s. if j \<in> {i} then x i else 0)"
unfolding indicator_fn_def using assms *
by (auto intro!: setsum_cong elim!: pos_simpleE)
show ?thesis unfolding ** setsum_cases[OF `finite s`] xi
using `i \<in> s` by simp
qed
lemma pos_simple_common_partition:
assumes psf: "(s, a, x) \<in> pos_simple f"
assumes psg: "(s', b, y) \<in> pos_simple g"
obtains z z' c k where "(k, c, z) \<in> pos_simple f" "(k, c, z') \<in> pos_simple g"
proof -
(* definitions *)
def k == "{0 ..< card (s \<times> s')}"
have fs: "finite s" "finite s'" "finite k" unfolding k_def
using psf psg unfolding pos_simple_def by auto
hence "finite (s \<times> s')" by simp
then obtain p where p: "p ` k = s \<times> s'" "inj_on p k"
using ex_bij_betw_nat_finite[of "s \<times> s'"] unfolding bij_betw_def k_def by blast
def c == "\<lambda> i. a (fst (p i)) \<inter> b (snd (p i))"
def z == "\<lambda> i. x (fst (p i))"
def z' == "\<lambda> i. y (snd (p i))"
have "finite k" unfolding k_def by simp
have "nonneg z" and "nonneg z'"
using psf psg unfolding z_def z'_def pos_simple_def nonneg_def by auto
have ck_subset_M: "c ` k \<subseteq> sets M"
proof
fix x assume "x \<in> c ` k"
then obtain j where "j \<in> k" and "x = c j" by auto
hence "p j \<in> s \<times> s'" using p(1) by auto
hence "a (fst (p j)) \<in> sets M" (is "?a \<in> _")
and "b (snd (p j)) \<in> sets M" (is "?b \<in> _")
using psf psg unfolding pos_simple_def by auto
thus "x \<in> sets M" unfolding `x = c j` c_def using Int by simp
qed
{ fix t assume "t \<in> space M"
hence ex1s: "\<exists>!i\<in>s. t \<in> a i" and ex1s': "\<exists>!i\<in>s'. t \<in> b i"
using psf psg unfolding pos_simple_def by auto
then obtain j j' where j: "j \<in> s" "t \<in> a j" and j': "j' \<in> s'" "t \<in> b j'"
by auto
then obtain i :: nat where i: "(j,j') = p i" and "i \<in> k" using p by auto
have "\<exists>!i\<in>k. t \<in> c i"
proof (rule ex1I[of _ i])
show "\<And>x. x \<in> k \<Longrightarrow> t \<in> c x \<Longrightarrow> x = i"
proof -
fix l assume "l \<in> k" "t \<in> c l"
hence "p l \<in> s \<times> s'" and t_in: "t \<in> a (fst (p l))" "t \<in> b (snd (p l))"
using p unfolding c_def by auto
hence "fst (p l) \<in> s" and "snd (p l) \<in> s'" by auto
with t_in j j' ex1s ex1s' have "p l = (j, j')" by (cases "p l", auto)
thus "l = i"
using `(j, j') = p i` p(2)[THEN inj_onD] `l \<in> k` `i \<in> k` by auto
qed
show "i \<in> k \<and> t \<in> c i"
using `i \<in> k` `t \<in> a j` `t \<in> b j'` c_def i[symmetric] by auto
qed auto
} note ex1 = this
show thesis
proof (rule that)
{ fix t i assume "t \<in> space M" and "i \<in> k"
hence "p i \<in> s \<times> s'" using p(1) by auto
hence "fst (p i) \<in> s" by auto
moreover
assume "t \<in> c i"
hence "t \<in> a (fst (p i))" unfolding c_def by auto
ultimately have "f t = z i" using psf `t \<in> space M`
unfolding z_def pos_simple_def by auto
}
thus "(k, c, z) \<in> pos_simple f"
using psf `finite k` `nonneg z` ck_subset_M ex1
unfolding pos_simple_def by auto
{ fix t i assume "t \<in> space M" and "i \<in> k"
hence "p i \<in> s \<times> s'" using p(1) by auto
hence "snd (p i) \<in> s'" by auto
moreover
assume "t \<in> c i"
hence "t \<in> b (snd (p i))" unfolding c_def by auto
ultimately have "g t = z' i" using psg `t \<in> space M`
unfolding z'_def pos_simple_def by auto
}
thus "(k, c, z') \<in> pos_simple g"
using psg `finite k` `nonneg z'` ck_subset_M ex1
unfolding pos_simple_def by auto
qed
qed
lemma pos_simple_integral_equal:
assumes psx: "(s, a, x) \<in> pos_simple f"
assumes psy: "(s', b, y) \<in> pos_simple f"
shows "pos_simple_integral (s, a, x) = pos_simple_integral (s', b, y)"
unfolding pos_simple_integral_def
proof simp
have "(\<Sum>i\<in>s. x i * measure M (a i)) =
(\<Sum>i\<in>s. (\<Sum>j \<in> s'. x i * measure M (a i \<inter> b j)))" (is "?left = _")
using psy psx unfolding setsum_right_distrib[symmetric]
by (auto intro!: setsum_cong measure_setsum_split elim!: pos_simpleE)
also have "... = (\<Sum>i\<in>s. (\<Sum>j \<in> s'. y j * measure M (a i \<inter> b j)))"
proof (rule setsum_cong, simp, rule setsum_cong, simp_all)
fix i j assume i: "i \<in> s" and j: "j \<in> s'"
hence "a i \<in> sets M" using psx by (auto elim!: pos_simpleE)
show "measure M (a i \<inter> b j) = 0 \<or> x i = y j"
proof (cases "a i \<inter> b j = {}")
case True thus ?thesis using empty_measure by simp
next
case False then obtain t where t: "t \<in> a i" "t \<in> b j" by auto
hence "t \<in> space M" using `a i \<in> sets M` sets_into_space by auto
with psx psy t i j have "x i = f t" and "y j = f t"
unfolding pos_simple_def by auto
thus ?thesis by simp
qed
qed
also have "... = (\<Sum>j\<in>s'. (\<Sum>i\<in>s. y j * measure M (a i \<inter> b j)))"
by (subst setsum_commute) simp
also have "... = (\<Sum>i\<in>s'. y i * measure M (b i))" (is "?sum_sum = ?right")
proof (rule setsum_cong)
fix j assume "j \<in> s'"
have "y j * measure M (b j) = (\<Sum>i\<in>s. y j * measure M (b j \<inter> a i))"
using psx psy `j \<in> s'` unfolding setsum_right_distrib[symmetric]
by (auto intro!: measure_setsum_split elim!: pos_simpleE)
thus "(\<Sum>i\<in>s. y j * measure M (a i \<inter> b j)) = y j * measure M (b j)"
by (auto intro!: setsum_cong arg_cong[where f="measure M"])
qed simp
finally show "?left = ?right" .
qed
lemma psfis_present:
assumes "A \<in> psfis f"
assumes "B \<in> psfis g"
obtains z z' c k where
"A = pos_simple_integral (k, c, z)"
"B = pos_simple_integral (k, c, z')"
"(k, c, z) \<in> pos_simple f"
"(k, c, z') \<in> pos_simple g"
using assms
proof -
from assms obtain s a x s' b y where
ps: "(s, a, x) \<in> pos_simple f" "(s', b, y) \<in> pos_simple g" and
A: "A = pos_simple_integral (s, a, x)" and
B: "B = pos_simple_integral (s', b, y)"
unfolding psfis_def pos_simple_integral_def by auto
guess z z' c k using pos_simple_common_partition[OF ps] . note part = this
show thesis
proof (rule that[of k c z z', OF _ _ part])
show "A = pos_simple_integral (k, c, z)"
unfolding A by (rule pos_simple_integral_equal[OF ps(1) part(1)])
show "B = pos_simple_integral (k, c, z')"
unfolding B by (rule pos_simple_integral_equal[OF ps(2) part(2)])
qed
qed
lemma pos_simple_integral_add:
assumes "(s, a, x) \<in> pos_simple f"
assumes "(s', b, y) \<in> pos_simple g"
obtains s'' c z where
"(s'', c, z) \<in> pos_simple (\<lambda>x. f x + g x)"
"(pos_simple_integral (s, a, x) +
pos_simple_integral (s', b, y) =
pos_simple_integral (s'', c, z))"
using assms
proof -
guess z z' c k
by (rule pos_simple_common_partition[OF `(s, a, x) \<in> pos_simple f ` `(s', b, y) \<in> pos_simple g`])
note kczz' = this
have x: "pos_simple_integral (s, a, x) = pos_simple_integral (k, c, z)"
by (rule pos_simple_integral_equal) (fact, fact)
have y: "pos_simple_integral (s', b, y) = pos_simple_integral (k, c, z')"
by (rule pos_simple_integral_equal) (fact, fact)
have "pos_simple_integral (k, c, (\<lambda> x. z x + z' x))
= (\<Sum> x \<in> k. (z x + z' x) * measure M (c x))"
unfolding pos_simple_integral_def by auto
also have "\<dots> = (\<Sum> x \<in> k. z x * measure M (c x) + z' x * measure M (c x))" using real_add_mult_distrib by auto
also have "\<dots> = (\<Sum> x \<in> k. z x * measure M (c x)) + (\<Sum> x \<in> k. z' x * measure M (c x))" using setsum_addf by auto
also have "\<dots> = pos_simple_integral (k, c, z) + pos_simple_integral (k, c, z')" unfolding pos_simple_integral_def by auto
finally have ths: "pos_simple_integral (s, a, x) + pos_simple_integral (s', b, y) =
pos_simple_integral (k, c, (\<lambda> x. z x + z' x))" using x y by auto
show ?thesis
apply (rule that[of k c "(\<lambda> x. z x + z' x)", OF _ ths])
using kczz' unfolding pos_simple_def nonneg_def by (auto intro!: add_nonneg_nonneg)
qed
lemma psfis_add:
assumes "a \<in> psfis f" "b \<in> psfis g"
shows "a + b \<in> psfis (\<lambda>x. f x + g x)"
using assms pos_simple_integral_add
unfolding psfis_def by auto blast
lemma pos_simple_integral_mono_on_mspace:
assumes f: "(s, a, x) \<in> pos_simple f"
assumes g: "(s', b, y) \<in> pos_simple g"
assumes mono: "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
shows "pos_simple_integral (s, a, x) \<le> pos_simple_integral (s', b, y)"
using assms
proof -
guess z z' c k by (rule pos_simple_common_partition[OF f g])
note kczz' = this
(* w = z and w' = z' except where c _ = {} or undef *)
def w == "\<lambda> i. if i \<notin> k \<or> c i = {} then 0 else z i"
def w' == "\<lambda> i. if i \<notin> k \<or> c i = {} then 0 else z' i"
{ fix i
have "w i \<le> w' i"
proof (cases "i \<notin> k \<or> c i = {}")
case False hence "i \<in> k" "c i \<noteq> {}" by auto
then obtain v where v: "v \<in> c i" and "c i \<in> sets M"
using kczz'(1) unfolding pos_simple_def by blast
hence "v \<in> space M" using sets_into_space by blast
with mono[OF `v \<in> space M`] kczz' `i \<in> k` `v \<in> c i`
have "z i \<le> z' i" unfolding pos_simple_def by simp
thus ?thesis unfolding w_def w'_def using False by auto
next
case True thus ?thesis unfolding w_def w'_def by auto
qed
} note w_mn = this
(* some technical stuff for the calculation*)
have "\<And> i. i \<in> k \<Longrightarrow> c i \<in> sets M" using kczz' unfolding pos_simple_def by auto
hence "\<And> i. i \<in> k \<Longrightarrow> measure M (c i) \<ge> 0" using positive by auto
hence w_mono: "\<And> i. i \<in> k \<Longrightarrow> w i * measure M (c i) \<le> w' i * measure M (c i)"
using mult_right_mono w_mn by blast
{ fix i have "\<lbrakk>i \<in> k ; z i \<noteq> w i\<rbrakk> \<Longrightarrow> measure M (c i) = 0"
unfolding w_def by (cases "c i = {}") auto }
hence zw: "\<And> i. i \<in> k \<Longrightarrow> z i * measure M (c i) = w i * measure M (c i)" by auto
{ fix i have "i \<in> k \<Longrightarrow> z i * measure M (c i) = w i * measure M (c i)"
unfolding w_def by (cases "c i = {}") simp_all }
note zw = this
{ fix i have "i \<in> k \<Longrightarrow> z' i * measure M (c i) = w' i * measure M (c i)"
unfolding w'_def by (cases "c i = {}") simp_all }
note z'w' = this
(* the calculation *)
have "pos_simple_integral (s, a, x) = pos_simple_integral (k, c, z)"
using f kczz'(1) by (rule pos_simple_integral_equal)
also have "\<dots> = (\<Sum> i \<in> k. z i * measure M (c i))"
unfolding pos_simple_integral_def by auto
also have "\<dots> = (\<Sum> i \<in> k. w i * measure M (c i))"
using setsum_cong2[of k, OF zw] by auto
also have "\<dots> \<le> (\<Sum> i \<in> k. w' i * measure M (c i))"
using setsum_mono[OF w_mono, of k] by auto
also have "\<dots> = (\<Sum> i \<in> k. z' i * measure M (c i))"
using setsum_cong2[of k, OF z'w'] by auto
also have "\<dots> = pos_simple_integral (k, c, z')"
unfolding pos_simple_integral_def by auto
also have "\<dots> = pos_simple_integral (s', b, y)"
using kczz'(2) g by (rule pos_simple_integral_equal)
finally show "pos_simple_integral (s, a, x) \<le> pos_simple_integral (s', b, y)"
by simp
qed
lemma pos_simple_integral_mono:
assumes a: "(s, a, x) \<in> pos_simple f" "(s', b, y) \<in> pos_simple g"
assumes "\<And> z. f z \<le> g z"
shows "pos_simple_integral (s, a, x) \<le> pos_simple_integral (s', b, y)"
using assms pos_simple_integral_mono_on_mspace by auto
lemma psfis_mono:
assumes "a \<in> psfis f" "b \<in> psfis g"
assumes "\<And> x. x \<in> space M \<Longrightarrow> f x \<le> g x"
shows "a \<le> b"
using assms pos_simple_integral_mono_on_mspace unfolding psfis_def by auto
lemma pos_simple_fn_integral_unique:
assumes "(s, a, x) \<in> pos_simple f" "(s', b, y) \<in> pos_simple f"
shows "pos_simple_integral (s, a, x) = pos_simple_integral (s', b, y)"
using assms real_le_antisym real_le_refl pos_simple_integral_mono by metis
lemma psfis_unique:
assumes "a \<in> psfis f" "b \<in> psfis f"
shows "a = b"
using assms real_le_antisym real_le_refl psfis_mono by metis
lemma pos_simple_integral_indicator:
assumes "A \<in> sets M"
obtains s a x where
"(s, a, x) \<in> pos_simple (indicator_fn A)"
"measure M A = pos_simple_integral (s, a, x)"
using assms
proof -
def s == "{0, 1} :: nat set"
def a == "\<lambda> i :: nat. if i = 0 then A else space M - A"
def x == "\<lambda> i :: nat. if i = 0 then 1 else (0 :: real)"
have eq: "pos_simple_integral (s, a, x) = measure M A"
unfolding s_def a_def x_def pos_simple_integral_def by auto
have "(s, a, x) \<in> pos_simple (indicator_fn A)"
unfolding pos_simple_def indicator_fn_def s_def a_def x_def nonneg_def
using assms sets_into_space by auto
from that[OF this] eq show thesis by auto
qed
lemma psfis_indicator:
assumes "A \<in> sets M"
shows "measure M A \<in> psfis (indicator_fn A)"
using pos_simple_integral_indicator[OF assms]
unfolding psfis_def image_def by auto
lemma pos_simple_integral_mult:
assumes f: "(s, a, x) \<in> pos_simple f"
assumes "0 \<le> z"
obtains s' b y where
"(s', b, y) \<in> pos_simple (\<lambda>x. z * f x)"
"pos_simple_integral (s', b, y) = z * pos_simple_integral (s, a, x)"
using assms that[of s a "\<lambda>i. z * x i"]
by (simp add: pos_simple_def pos_simple_integral_def setsum_right_distrib algebra_simps nonneg_def mult_nonneg_nonneg)
lemma psfis_mult:
assumes "r \<in> psfis f"
assumes "0 \<le> z"
shows "z * r \<in> psfis (\<lambda>x. z * f x)"
using assms
proof -
from assms obtain s a x
where sax: "(s, a, x) \<in> pos_simple f"
and r: "r = pos_simple_integral (s, a, x)"
unfolding psfis_def image_def by auto
obtain s' b y where
"(s', b, y) \<in> pos_simple (\<lambda>x. z * f x)"
"z * pos_simple_integral (s, a, x) = pos_simple_integral (s', b, y)"
using pos_simple_integral_mult[OF sax `0 \<le> z`, of thesis] by auto
thus ?thesis using r unfolding psfis_def image_def by auto
qed
lemma psfis_setsum_image:
assumes "finite P"
assumes "\<And>i. i \<in> P \<Longrightarrow> a i \<in> psfis (f i)"
shows "(setsum a P) \<in> psfis (\<lambda>t. \<Sum>i \<in> P. f i t)"
using assms
proof (induct P)
case empty
let ?s = "{0 :: nat}"
let ?a = "\<lambda> i. if i = (0 :: nat) then space M else {}"
let ?x = "\<lambda> (i :: nat). (0 :: real)"
have "(?s, ?a, ?x) \<in> pos_simple (\<lambda> t. (0 :: real))"
unfolding pos_simple_def image_def nonneg_def by auto
moreover have "(\<Sum> i \<in> ?s. ?x i * measure M (?a i)) = 0" by auto
ultimately have "0 \<in> psfis (\<lambda> t. 0)"
unfolding psfis_def image_def pos_simple_integral_def nonneg_def
by (auto intro!:bexI[of _ "(?s, ?a, ?x)"])
thus ?case by auto
next
case (insert x P) note asms = this
have "finite P" by fact
have "x \<notin> P" by fact
have "(\<And>i. i \<in> P \<Longrightarrow> a i \<in> psfis (f i)) \<Longrightarrow>
setsum a P \<in> psfis (\<lambda>t. \<Sum>i\<in>P. f i t)" by fact
have "setsum a (insert x P) = a x + setsum a P" using `finite P` `x \<notin> P` by auto
also have "\<dots> \<in> psfis (\<lambda> t. f x t + (\<Sum> i \<in> P. f i t))"
using asms psfis_add by auto
also have "\<dots> = psfis (\<lambda> t. \<Sum> i \<in> insert x P. f i t)"
using `x \<notin> P` `finite P` by auto
finally show ?case by simp
qed
lemma psfis_intro:
assumes "a ` P \<subseteq> sets M" and "nonneg x" and "finite P"
shows "(\<Sum>i\<in>P. x i * measure M (a i)) \<in> psfis (\<lambda>t. \<Sum>i\<in>P. x i * indicator_fn (a i) t)"
using assms psfis_mult psfis_indicator
unfolding image_def nonneg_def
by (auto intro!:psfis_setsum_image)
lemma psfis_nonneg: "a \<in> psfis f \<Longrightarrow> nonneg f"
unfolding psfis_def pos_simple_def by auto
lemma pos_psfis: "r \<in> psfis f \<Longrightarrow> 0 \<le> r"
unfolding psfis_def pos_simple_integral_def image_def pos_simple_def nonneg_def
using positive[unfolded positive_def] by (auto intro!:setsum_nonneg mult_nonneg_nonneg)
lemma psfis_borel_measurable:
assumes "a \<in> psfis f"
shows "f \<in> borel_measurable M"
using assms
proof -
from assms obtain s a' x where sa'x: "(s, a', x) \<in> pos_simple f" and sa'xa: "pos_simple_integral (s, a', x) = a"
and fs: "finite s"
unfolding psfis_def pos_simple_integral_def image_def
by (auto elim:pos_simpleE)
{ fix w assume "w \<in> space M"
hence "(f w \<le> a) = ((\<Sum> i \<in> s. x i * indicator_fn (a' i) w) \<le> a)"
using pos_simple_setsum_indicator_fn[OF sa'x, of w] by simp
} hence "\<And> w. (w \<in> space M \<and> f w \<le> a) = (w \<in> space M \<and> (\<Sum> i \<in> s. x i * indicator_fn (a' i) w) \<le> a)"
by auto
{ fix i assume "i \<in> s"
hence "indicator_fn (a' i) \<in> borel_measurable M"
using borel_measurable_indicator using sa'x[unfolded pos_simple_def] by auto
hence "(\<lambda> w. x i * indicator_fn (a' i) w) \<in> borel_measurable M"
using affine_borel_measurable[of "\<lambda> w. indicator_fn (a' i) w" 0 "x i"]
real_mult_commute by auto }
from borel_measurable_setsum_borel_measurable[OF fs this] affine_borel_measurable
have "(\<lambda> w. (\<Sum> i \<in> s. x i * indicator_fn (a' i) w)) \<in> borel_measurable M" by auto
from borel_measurable_cong[OF pos_simple_setsum_indicator_fn[OF sa'x]] this
show ?thesis by simp
qed
lemma psfis_mono_conv_mono:
assumes mono: "mono_convergent u f (space M)"
and ps_u: "\<And>n. x n \<in> psfis (u n)"
and "x ----> y"
and "r \<in> psfis s"
and f_upper_bound: "\<And>t. t \<in> space M \<Longrightarrow> s t \<le> f t"
shows "r <= y"
proof (rule field_le_mult_one_interval)
fix z :: real assume "0 < z" and "z < 1"
hence "0 \<le> z" by auto
let "?B' n" = "{w \<in> space M. z * s w <= u n w}"
have "incseq x" unfolding incseq_def
proof safe
fix m n :: nat assume "m \<le> n"
show "x m \<le> x n"
proof (rule psfis_mono[OF `x m \<in> psfis (u m)` `x n \<in> psfis (u n)`])
fix t assume "t \<in> space M"
with mono_convergentD[OF mono this] `m \<le> n` show "u m t \<le> u n t"
unfolding incseq_def by auto
qed
qed
from `r \<in> psfis s`
obtain s' a x' where r: "r = pos_simple_integral (s', a, x')"
and ps_s: "(s', a, x') \<in> pos_simple s"
unfolding psfis_def by auto
{ fix t i assume "i \<in> s'" "t \<in> a i"
hence "t \<in> space M" using ps_s by (auto elim!: pos_simpleE) }
note t_in_space = this
{ fix n
from psfis_borel_measurable[OF `r \<in> psfis s`]
psfis_borel_measurable[OF ps_u]
have "?B' n \<in> sets M"
by (auto intro!:
borel_measurable_leq_borel_measurable
borel_measurable_times_borel_measurable
borel_measurable_const) }
note B'_in_M = this
{ fix n have "(\<lambda>i. a i \<inter> ?B' n) ` s' \<subseteq> sets M" using B'_in_M ps_s
by (auto intro!: Int elim!: pos_simpleE) }
note B'_inter_a_in_M = this
let "?sum n" = "(\<Sum>i\<in>s'. x' i * measure M (a i \<inter> ?B' n))"
{ fix n
have "z * ?sum n \<le> x n"
proof (rule psfis_mono[OF _ ps_u])
have *: "\<And>i t. indicator_fn (?B' n) t * (x' i * indicator_fn (a i) t) =
x' i * indicator_fn (a i \<inter> ?B' n) t" unfolding indicator_fn_def by auto
have ps': "?sum n \<in> psfis (\<lambda>t. indicator_fn (?B' n) t * (\<Sum>i\<in>s'. x' i * indicator_fn (a i) t))"
unfolding setsum_right_distrib * using B'_in_M ps_s
by (auto intro!: psfis_intro Int elim!: pos_simpleE)
also have "... = psfis (\<lambda>t. indicator_fn (?B' n) t * s t)" (is "psfis ?l = psfis ?r")
proof (rule psfis_cong)
show "nonneg ?l" using psfis_nonneg[OF ps'] .
show "nonneg ?r" using psfis_nonneg[OF `r \<in> psfis s`] unfolding nonneg_def indicator_fn_def by auto
fix t assume "t \<in> space M"
show "?l t = ?r t" unfolding pos_simple_setsum_indicator_fn[OF ps_s `t \<in> space M`] ..
qed
finally show "z * ?sum n \<in> psfis (\<lambda>t. z * ?r t)" using psfis_mult[OF _ `0 \<le> z`] by simp
next
fix t assume "t \<in> space M"
show "z * (indicator_fn (?B' n) t * s t) \<le> u n t"
using psfis_nonneg[OF ps_u] unfolding nonneg_def indicator_fn_def by auto
qed }
hence *: "\<exists>N. \<forall>n\<ge>N. z * ?sum n \<le> x n" by (auto intro!: exI[of _ 0])
show "z * r \<le> y" unfolding r pos_simple_integral_def
proof (rule LIMSEQ_le[OF mult_right.LIMSEQ `x ----> y` *],
simp, rule LIMSEQ_setsum, rule mult_right.LIMSEQ)
fix i assume "i \<in> s'"
from psfis_nonneg[OF `r \<in> psfis s`, unfolded nonneg_def]
have "\<And>t. 0 \<le> s t" by simp
have *: "(\<Union>j. a i \<inter> ?B' j) = a i"
proof (safe, simp, safe)
fix t assume "t \<in> a i"
thus "t \<in> space M" using t_in_space[OF `i \<in> s'`] by auto
show "\<exists>j. z * s t \<le> u j t"
proof (cases "s t = 0")
case True thus ?thesis
using psfis_nonneg[OF ps_u] unfolding nonneg_def by auto
next
case False with `0 \<le> s t`
have "0 < s t" by auto
hence "z * s t < 1 * s t" using `0 < z` `z < 1`
by (auto intro!: mult_strict_right_mono simp del: mult_1_left)
also have "... \<le> f t" using f_upper_bound `t \<in> space M` by auto
finally obtain b where "\<And>j. b \<le> j \<Longrightarrow> z * s t < u j t" using `t \<in> space M`
using mono_conv_outgrow[of "\<lambda>n. u n t" "f t" "z * s t"]
using mono_convergentD[OF mono] by auto
from this[of b] show ?thesis by (auto intro!: exI[of _ b])
qed
qed
show "(\<lambda>n. measure M (a i \<inter> ?B' n)) ----> measure M (a i)"
proof (safe intro!:
monotone_convergence[of "\<lambda>n. a i \<inter> ?B' n", unfolded comp_def *])
fix n show "a i \<inter> ?B' n \<in> sets M"
using B'_inter_a_in_M[of n] `i \<in> s'` by auto
next
fix j t assume "t \<in> space M" and "z * s t \<le> u j t"
thus "z * s t \<le> u (Suc j) t"
using mono_convergentD(1)[OF mono, unfolded incseq_def,
rule_format, of t j "Suc j"]
by auto
qed
qed
qed
section "Continuous posititve integration"
definition
"nnfis f = { y. \<exists>u x. mono_convergent u f (space M) \<and>
(\<forall>n. x n \<in> psfis (u n)) \<and> x ----> y }"
lemma psfis_nnfis:
"a \<in> psfis f \<Longrightarrow> a \<in> nnfis f"
unfolding nnfis_def mono_convergent_def incseq_def
by (auto intro!: exI[of _ "\<lambda>n. f"] exI[of _ "\<lambda>n. a"] LIMSEQ_const)
lemma nnfis_0: "0 \<in> nnfis (\<lambda> x. 0)"
by (rule psfis_nnfis[OF psfis_0])
lemma nnfis_times:
assumes "a \<in> nnfis f" and "0 \<le> z"
shows "z * a \<in> nnfis (\<lambda>t. z * f t)"
proof -
obtain u x where "mono_convergent u f (space M)" and
"\<And>n. x n \<in> psfis (u n)" "x ----> a"
using `a \<in> nnfis f` unfolding nnfis_def by auto
with `0 \<le> z`show ?thesis unfolding nnfis_def mono_convergent_def incseq_def
by (auto intro!: exI[of _ "\<lambda>n w. z * u n w"] exI[of _ "\<lambda>n. z * x n"]
LIMSEQ_mult LIMSEQ_const psfis_mult mult_mono1)
qed
lemma nnfis_add:
assumes "a \<in> nnfis f" and "b \<in> nnfis g"
shows "a + b \<in> nnfis (\<lambda>t. f t + g t)"
proof -
obtain u x w y
where "mono_convergent u f (space M)" and
"\<And>n. x n \<in> psfis (u n)" "x ----> a" and
"mono_convergent w g (space M)" and
"\<And>n. y n \<in> psfis (w n)" "y ----> b"
using `a \<in> nnfis f` `b \<in> nnfis g` unfolding nnfis_def by auto
thus ?thesis unfolding nnfis_def mono_convergent_def incseq_def
by (auto intro!: exI[of _ "\<lambda>n a. u n a + w n a"] exI[of _ "\<lambda>n. x n + y n"]
LIMSEQ_add LIMSEQ_const psfis_add add_mono)
qed
lemma nnfis_mono:
assumes nnfis: "a \<in> nnfis f" "b \<in> nnfis g"
and mono: "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
shows "a \<le> b"
proof -
obtain u x w y where
mc: "mono_convergent u f (space M)" "mono_convergent w g (space M)" and
psfis: "\<And>n. x n \<in> psfis (u n)" "\<And>n. y n \<in> psfis (w n)" and
limseq: "x ----> a" "y ----> b" using nnfis unfolding nnfis_def by auto
show ?thesis
proof (rule LIMSEQ_le_const2[OF limseq(1)], rule exI[of _ 0], safe)
fix n
show "x n \<le> b"
proof (rule psfis_mono_conv_mono[OF mc(2) psfis(2) limseq(2) psfis(1)])
fix t assume "t \<in> space M"
from mono_convergent_le[OF mc(1) this] mono[OF this]
show "u n t \<le> g t" by (rule order_trans)
qed
qed
qed
lemma nnfis_unique:
assumes a: "a \<in> nnfis f" and b: "b \<in> nnfis f"
shows "a = b"
using nnfis_mono[OF a b] nnfis_mono[OF b a]
by (auto intro!: real_le_antisym[of a b])
lemma psfis_equiv:
assumes "a \<in> psfis f" and "nonneg g"
and "\<And>t. t \<in> space M \<Longrightarrow> f t = g t"
shows "a \<in> psfis g"
using assms unfolding psfis_def pos_simple_def by auto
lemma psfis_mon_upclose:
assumes "\<And>m. a m \<in> psfis (u m)"
shows "\<exists>c. c \<in> psfis (mon_upclose n u)"
proof (induct n)
case 0 thus ?case unfolding mon_upclose.simps using assms ..
next
case (Suc n)
then obtain sn an xn where ps: "(sn, an, xn) \<in> pos_simple (mon_upclose n u)"
unfolding psfis_def by auto
obtain ss as xs where ps': "(ss, as, xs) \<in> pos_simple (u (Suc n))"
using assms[of "Suc n"] unfolding psfis_def by auto
from pos_simple_common_partition[OF ps ps'] guess x x' a s .
hence "(s, a, upclose x x') \<in> pos_simple (mon_upclose (Suc n) u)"
by (simp add: upclose_def pos_simple_def nonneg_def max_def)
thus ?case unfolding psfis_def by auto
qed
text {* Beppo-Levi monotone convergence theorem *}
lemma nnfis_mon_conv:
assumes mc: "mono_convergent f h (space M)"
and nnfis: "\<And>n. x n \<in> nnfis (f n)"
and "x ----> z"
shows "z \<in> nnfis h"
proof -
have "\<forall>n. \<exists>u y. mono_convergent u (f n) (space M) \<and> (\<forall>m. y m \<in> psfis (u m)) \<and>
y ----> x n"
using nnfis unfolding nnfis_def by auto
from choice[OF this] guess u ..
from choice[OF this] guess y ..
hence mc_u: "\<And>n. mono_convergent (u n) (f n) (space M)"
and psfis: "\<And>n m. y n m \<in> psfis (u n m)" and "\<And>n. y n ----> x n"
by auto
let "?upclose n" = "mon_upclose n (\<lambda>m. u m n)"
have "\<exists>c. \<forall>n. c n \<in> psfis (?upclose n)"
by (safe intro!: choice psfis_mon_upclose) (rule psfis)
then guess c .. note c = this[rule_format]
show ?thesis unfolding nnfis_def
proof (safe intro!: exI)
show mc_upclose: "mono_convergent ?upclose h (space M)"
by (rule mon_upclose_mono_convergent[OF mc_u mc])
show psfis_upclose: "\<And>n. c n \<in> psfis (?upclose n)"
using c .
{ fix n m :: nat assume "n \<le> m"
hence "c n \<le> c m"
using psfis_mono[OF c c]
using mono_convergentD(1)[OF mc_upclose, unfolded incseq_def]
by auto }
hence "incseq c" unfolding incseq_def by auto
{ fix n
have c_nnfis: "c n \<in> nnfis (?upclose n)" using c by (rule psfis_nnfis)
from nnfis_mono[OF c_nnfis nnfis]
mon_upclose_le_mono_convergent[OF mc_u]
mono_convergentD(1)[OF mc]
have "c n \<le> x n" by fast }
note c_less_x = this
{ fix n
note c_less_x[of n]
also have "x n \<le> z"
proof (rule incseq_le)
show "x ----> z" by fact
from mono_convergentD(1)[OF mc]
show "incseq x" unfolding incseq_def
by (auto intro!: nnfis_mono[OF nnfis nnfis])
qed
finally have "c n \<le> z" . }
note c_less_z = this
have "convergent c"
proof (rule Bseq_mono_convergent[unfolded incseq_def[symmetric]])
show "Bseq c"
using pos_psfis[OF c] c_less_z
by (auto intro!: BseqI'[of _ z])
show "incseq c" by fact
qed
then obtain l where l: "c ----> l" unfolding convergent_def by auto
have "l \<le> z" using c_less_x l
by (auto intro!: LIMSEQ_le[OF _ `x ----> z`])
moreover have "z \<le> l"
proof (rule LIMSEQ_le_const2[OF `x ----> z`], safe intro!: exI[of _ 0])
fix n
have "l \<in> nnfis h"
unfolding nnfis_def using l mc_upclose psfis_upclose by auto
from nnfis this mono_convergent_le[OF mc]
show "x n \<le> l" by (rule nnfis_mono)
qed
ultimately have "l = z" by (rule real_le_antisym)
thus "c ----> z" using `c ----> l` by simp
qed
qed
lemma nnfis_pos_on_mspace:
assumes "a \<in> nnfis f" and "x \<in>space M"
shows "0 \<le> f x"
using assms
proof -
from assms[unfolded nnfis_def] guess u y by auto note uy = this
hence "\<And> n. 0 \<le> u n x"
unfolding nnfis_def psfis_def pos_simple_def nonneg_def mono_convergent_def
by auto
thus "0 \<le> f x" using uy[rule_format]
unfolding nnfis_def psfis_def pos_simple_def nonneg_def mono_convergent_def
using incseq_le[of "\<lambda> n. u n x" "f x"] real_le_trans
by fast
qed
lemma nnfis_borel_measurable:
assumes "a \<in> nnfis f"
shows "f \<in> borel_measurable M"
using assms unfolding nnfis_def
apply auto
apply (rule mono_convergent_borel_measurable)
using psfis_borel_measurable
by auto
lemma borel_measurable_mon_conv_psfis:
assumes f_borel: "f \<in> borel_measurable M"
and nonneg: "\<And>t. t \<in> space M \<Longrightarrow> 0 \<le> f t"
shows"\<exists>u x. mono_convergent u f (space M) \<and> (\<forall>n. x n \<in> psfis (u n))"
proof (safe intro!: exI)
let "?I n" = "{0<..<n * 2^n}"
let "?A n i" = "{w \<in> space M. real (i :: nat) / 2^(n::nat) \<le> f w \<and> f w < real (i + 1) / 2^n}"
let "?u n t" = "\<Sum>i\<in>?I n. real i / 2^n * indicator_fn (?A n i) t"
let "?x n" = "\<Sum>i\<in>?I n. real i / 2^n * measure M (?A n i)"
let "?w n t" = "if f t < real n then real (natfloor (f t * 2^n)) / 2^n else 0"
{ fix t n assume t: "t \<in> space M"
have "?u n t = ?w n t" (is "_ = (if _ then real ?i / _ else _)")
proof (cases "f t < real n")
case True
with nonneg t
have i: "?i < n * 2^n"
by (auto simp: real_of_nat_power[symmetric]
intro!: less_natfloor mult_nonneg_nonneg)
hence t_in_A: "t \<in> ?A n ?i"
unfolding divide_le_eq less_divide_eq
using nonneg t True
by (auto intro!: real_natfloor_le
real_natfloor_gt_diff_one[unfolded diff_less_eq]
simp: real_of_nat_Suc zero_le_mult_iff)
hence *: "real ?i / 2^n \<le> f t"
"f t < real (?i + 1) / 2^n" by auto
{ fix j assume "t \<in> ?A n j"
hence "real j / 2^n \<le> f t"
and "f t < real (j + 1) / 2^n" by auto
with * have "j \<in> {?i}" unfolding divide_le_eq less_divide_eq
by auto }
hence *: "\<And>j. t \<in> ?A n j \<longleftrightarrow> j \<in> {?i}" using t_in_A by auto
have "?u n t = real ?i / 2^n"
unfolding indicator_fn_def if_distrib *
setsum_cases[OF finite_greaterThanLessThan]
using i by (cases "?i = 0") simp_all
thus ?thesis using True by auto
next
case False
have "?u n t = (\<Sum>i \<in> {0 <..< n*2^n}. 0)"
proof (rule setsum_cong)
fix i assume "i \<in> {0 <..< n*2^n}"
hence "i + 1 \<le> n * 2^n" by simp
hence "real (i + 1) \<le> real n * 2^n"
unfolding real_of_nat_le_iff[symmetric]
by (auto simp: real_of_nat_power[symmetric])
also have "... \<le> f t * 2^n"
using False by (auto intro!: mult_nonneg_nonneg)
finally have "t \<notin> ?A n i"
by (auto simp: divide_le_eq less_divide_eq)
thus "real i / 2^n * indicator_fn (?A n i) t = 0"
unfolding indicator_fn_def by auto
qed simp
thus ?thesis using False by auto
qed }
note u_at_t = this
show "mono_convergent ?u f (space M)" unfolding mono_convergent_def
proof safe
fix t assume t: "t \<in> space M"
{ fix m n :: nat assume "m \<le> n"
hence *: "(2::real)^n = 2^m * 2^(n - m)" unfolding class_semiring.mul_pwr by auto
have "real (natfloor (f t * 2^m) * natfloor (2^(n-m))) \<le> real (natfloor (f t * 2 ^ n))"
apply (subst *)
apply (subst class_semiring.mul_a)
apply (subst real_of_nat_le_iff)
apply (rule le_mult_natfloor)
using nonneg[OF t] by (auto intro!: mult_nonneg_nonneg)
hence "real (natfloor (f t * 2^m)) * 2^n \<le> real (natfloor (f t * 2^n)) * 2^m"
apply (subst *)
apply (subst (3) class_semiring.mul_c)
apply (subst class_semiring.mul_a)
by (auto intro: mult_right_mono simp: natfloor_power real_of_nat_power[symmetric]) }
thus "incseq (\<lambda>n. ?u n t)" unfolding u_at_t[OF t] unfolding incseq_def
by (auto simp add: le_divide_eq divide_le_eq less_divide_eq)
show "(\<lambda>i. ?u i t) ----> f t" unfolding u_at_t[OF t]
proof (rule LIMSEQ_I, safe intro!: exI)
fix r :: real and n :: nat
let ?N = "natfloor (1/r) + 1"
assume "0 < r" and N: "max ?N (natfloor (f t) + 1) \<le> n"
hence "?N \<le> n" by auto
have "1 / r < real (natfloor (1/r) + 1)" using real_natfloor_add_one_gt
by (simp add: real_of_nat_Suc)
also have "... < 2^?N" by (rule two_realpow_gt)
finally have less_r: "1 / 2^?N < r" using `0 < r`
by (auto simp: less_divide_eq divide_less_eq algebra_simps)
have "f t < real (natfloor (f t) + 1)" using real_natfloor_add_one_gt[of "f t"] by auto
also have "... \<le> real n" unfolding real_of_nat_le_iff using N by auto
finally have "f t < real n" .
have "real (natfloor (f t * 2^n)) \<le> f t * 2^n"
using nonneg[OF t] by (auto intro!: real_natfloor_le mult_nonneg_nonneg)
hence less: "real (natfloor (f t * 2^n)) / 2^n \<le> f t" unfolding divide_le_eq by auto
have "f t * 2 ^ n - 1 < real (natfloor (f t * 2^n))" using real_natfloor_gt_diff_one .
hence "f t - real (natfloor (f t * 2^n)) / 2^n < 1 / 2^n"
by (auto simp: less_divide_eq divide_less_eq algebra_simps)
also have "... \<le> 1 / 2^?N" using `?N \<le> n`
by (auto intro!: divide_left_mono mult_pos_pos simp del: power_Suc)
also have "... < r" using less_r .
finally show "norm (?w n t - f t) < r" using `f t < real n` less by auto
qed
qed
fix n
show "?x n \<in> psfis (?u n)"
proof (rule psfis_intro)
show "?A n ` ?I n \<subseteq> sets M"
proof safe
fix i :: nat
from Int[OF
f_borel[unfolded borel_measurable_less_iff, rule_format, of "real (i+1) / 2^n"]
f_borel[unfolded borel_measurable_ge_iff, rule_format, of "real i / 2^n"]]
show "?A n i \<in> sets M"
by (metis Collect_conj_eq Int_commute Int_left_absorb Int_left_commute)
qed
show "nonneg (\<lambda>i :: nat. real i / 2 ^ n)"
unfolding nonneg_def by (auto intro!: divide_nonneg_pos)
qed auto
qed
lemma nnfis_dom_conv:
assumes borel: "f \<in> borel_measurable M"
and nnfis: "b \<in> nnfis g"
and ord: "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
and nonneg: "\<And>t. t \<in> space M \<Longrightarrow> 0 \<le> f t"
shows "\<exists>a. a \<in> nnfis f \<and> a \<le> b"
proof -
obtain u x where mc_f: "mono_convergent u f (space M)" and
psfis: "\<And>n. x n \<in> psfis (u n)"
using borel_measurable_mon_conv_psfis[OF borel nonneg] by auto
{ fix n
{ fix t assume t: "t \<in> space M"
note mono_convergent_le[OF mc_f this, of n]
also note ord[OF t]
finally have "u n t \<le> g t" . }
from nnfis_mono[OF psfis_nnfis[OF psfis] nnfis this]
have "x n \<le> b" by simp }
note x_less_b = this
have "convergent x"
proof (safe intro!: Bseq_mono_convergent)
from x_less_b pos_psfis[OF psfis]
show "Bseq x" by (auto intro!: BseqI'[of _ b])
fix n m :: nat assume *: "n \<le> m"
show "x n \<le> x m"
proof (rule psfis_mono[OF `x n \<in> psfis (u n)` `x m \<in> psfis (u m)`])
fix t assume "t \<in> space M"
from mc_f[THEN mono_convergentD(1), unfolded incseq_def, OF this]
show "u n t \<le> u m t" using * by auto
qed
qed
then obtain a where "x ----> a" unfolding convergent_def by auto
with LIMSEQ_le_const2[OF `x ----> a`] x_less_b mc_f psfis
show ?thesis unfolding nnfis_def by auto
qed
lemma the_nnfis[simp]: "a \<in> nnfis f \<Longrightarrow> (THE a. a \<in> nnfis f) = a"
by (auto intro: the_equality nnfis_unique)
lemma nnfis_cong:
assumes cong: "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
shows "nnfis f = nnfis g"
proof -
{ fix f g :: "'a \<Rightarrow> real" assume cong: "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
fix x assume "x \<in> nnfis f"
then guess u y unfolding nnfis_def by safe note x = this
hence "mono_convergent u g (space M)"
unfolding mono_convergent_def using cong by auto
with x(2,3) have "x \<in> nnfis g" unfolding nnfis_def by auto }
from this[OF cong] this[OF cong[symmetric]]
show ?thesis by safe
qed
section "Lebesgue Integral"
definition
"integrable f \<longleftrightarrow> (\<exists>x. x \<in> nnfis (pos_part f)) \<and> (\<exists>y. y \<in> nnfis (neg_part f))"
definition
"integral f = (THE i :: real. i \<in> nnfis (pos_part f)) - (THE j. j \<in> nnfis (neg_part f))"
lemma integral_cong:
assumes cong: "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"
shows "integral f = integral g"
proof -
have "nnfis (pos_part f) = nnfis (pos_part g)"
using cong by (auto simp: pos_part_def intro!: nnfis_cong)
moreover
have "nnfis (neg_part f) = nnfis (neg_part g)"
using cong by (auto simp: neg_part_def intro!: nnfis_cong)
ultimately show ?thesis
unfolding integral_def by auto
qed
lemma nnfis_integral:
assumes "a \<in> nnfis f"
shows "integrable f" and "integral f = a"
proof -
have a: "a \<in> nnfis (pos_part f)"
using assms nnfis_pos_on_mspace[OF assms]
by (auto intro!: nnfis_mon_conv[of "\<lambda>i. f" _ "\<lambda>i. a"]
LIMSEQ_const simp: mono_convergent_def pos_part_def incseq_def max_def)
have "\<And>t. t \<in> space M \<Longrightarrow> neg_part f t = 0"
unfolding neg_part_def min_def
using nnfis_pos_on_mspace[OF assms] by auto
hence 0: "0 \<in> nnfis (neg_part f)"
by (auto simp: nnfis_def mono_convergent_def psfis_0 incseq_def
intro!: LIMSEQ_const exI[of _ "\<lambda> x n. 0"] exI[of _ "\<lambda> n. 0"])
from 0 a show "integrable f" "integral f = a"
unfolding integrable_def integral_def by auto
qed
lemma nnfis_minus_nnfis_integral:
assumes "a \<in> nnfis f" and "b \<in> nnfis g"
shows "integrable (\<lambda>t. f t - g t)" and "integral (\<lambda>t. f t - g t) = a - b"
proof -
have borel: "(\<lambda>t. f t - g t) \<in> borel_measurable M" using assms
by (blast intro:
borel_measurable_diff_borel_measurable nnfis_borel_measurable)
have "\<exists>x. x \<in> nnfis (pos_part (\<lambda>t. f t - g t)) \<and> x \<le> a + b"
(is "\<exists>x. x \<in> nnfis ?pp \<and> _")
proof (rule nnfis_dom_conv)
show "?pp \<in> borel_measurable M"
using borel by (rule pos_part_borel_measurable neg_part_borel_measurable)
show "a + b \<in> nnfis (\<lambda>t. f t + g t)" using assms by (rule nnfis_add)
fix t assume "t \<in> space M"
with assms nnfis_add assms[THEN nnfis_pos_on_mspace[OF _ this]]
show "?pp t \<le> f t + g t" unfolding pos_part_def by auto
show "0 \<le> ?pp t" using nonneg_pos_part[of "\<lambda>t. f t - g t"]
unfolding nonneg_def by auto
qed
then obtain x where x: "x \<in> nnfis ?pp" by auto
moreover
have "\<exists>x. x \<in> nnfis (neg_part (\<lambda>t. f t - g t)) \<and> x \<le> a + b"
(is "\<exists>x. x \<in> nnfis ?np \<and> _")
proof (rule nnfis_dom_conv)
show "?np \<in> borel_measurable M"
using borel by (rule pos_part_borel_measurable neg_part_borel_measurable)
show "a + b \<in> nnfis (\<lambda>t. f t + g t)" using assms by (rule nnfis_add)
fix t assume "t \<in> space M"
with assms nnfis_add assms[THEN nnfis_pos_on_mspace[OF _ this]]
show "?np t \<le> f t + g t" unfolding neg_part_def by auto
show "0 \<le> ?np t" using nonneg_neg_part[of "\<lambda>t. f t - g t"]
unfolding nonneg_def by auto
qed
then obtain y where y: "y \<in> nnfis ?np" by auto
ultimately show "integrable (\<lambda>t. f t - g t)"
unfolding integrable_def by auto
from x and y
have "a + y \<in> nnfis (\<lambda>t. f t + ?np t)"
and "b + x \<in> nnfis (\<lambda>t. g t + ?pp t)" using assms by (auto intro: nnfis_add)
moreover
have "\<And>t. f t + ?np t = g t + ?pp t"
unfolding pos_part_def neg_part_def by auto
ultimately have "a - b = x - y"
using nnfis_unique by (auto simp: algebra_simps)
thus "integral (\<lambda>t. f t - g t) = a - b"
unfolding integral_def
using the_nnfis[OF x] the_nnfis[OF y] by simp
qed
lemma integral_borel_measurable:
"integrable f \<Longrightarrow> f \<in> borel_measurable M"
unfolding integrable_def
by (subst pos_part_neg_part_borel_measurable_iff)
(auto intro: nnfis_borel_measurable)
lemma integral_indicator_fn:
assumes "a \<in> sets M"
shows "integral (indicator_fn a) = measure M a"
and "integrable (indicator_fn a)"
using psfis_indicator[OF assms, THEN psfis_nnfis]
by (auto intro!: nnfis_integral)
lemma integral_add:
assumes "integrable f" and "integrable g"
shows "integrable (\<lambda>t. f t + g t)"
and "integral (\<lambda>t. f t + g t) = integral f + integral g"
proof -
{ fix t
have "pos_part f t + pos_part g t - (neg_part f t + neg_part g t) =
f t + g t"
unfolding pos_part_def neg_part_def by auto }
note part_sum = this
from assms obtain a b c d where
a: "a \<in> nnfis (pos_part f)" and b: "b \<in> nnfis (neg_part f)" and
c: "c \<in> nnfis (pos_part g)" and d: "d \<in> nnfis (neg_part g)"
unfolding integrable_def by auto
note sums = nnfis_add[OF a c] nnfis_add[OF b d]
note int = nnfis_minus_nnfis_integral[OF sums, unfolded part_sum]
show "integrable (\<lambda>t. f t + g t)" using int(1) .
show "integral (\<lambda>t. f t + g t) = integral f + integral g"
using int(2) sums a b c d by (simp add: the_nnfis integral_def)
qed
lemma integral_mono:
assumes "integrable f" and "integrable g"
and mono: "\<And>t. t \<in> space M \<Longrightarrow> f t \<le> g t"
shows "integral f \<le> integral g"
proof -
from assms obtain a b c d where
a: "a \<in> nnfis (pos_part f)" and b: "b \<in> nnfis (neg_part f)" and
c: "c \<in> nnfis (pos_part g)" and d: "d \<in> nnfis (neg_part g)"
unfolding integrable_def by auto
have "a \<le> c"
proof (rule nnfis_mono[OF a c])
fix t assume "t \<in> space M"
from mono[OF this] show "pos_part f t \<le> pos_part g t"
unfolding pos_part_def by auto
qed
moreover have "d \<le> b"
proof (rule nnfis_mono[OF d b])
fix t assume "t \<in> space M"
from mono[OF this] show "neg_part g t \<le> neg_part f t"
unfolding neg_part_def by auto
qed
ultimately have "a - b \<le> c - d" by auto
thus ?thesis unfolding integral_def
using a b c d by (simp add: the_nnfis)
qed
lemma integral_uminus:
assumes "integrable f"
shows "integrable (\<lambda>t. - f t)"
and "integral (\<lambda>t. - f t) = - integral f"
proof -
have "pos_part f = neg_part (\<lambda>t.-f t)" and "neg_part f = pos_part (\<lambda>t.-f t)"
unfolding pos_part_def neg_part_def by (auto intro!: ext)
with assms show "integrable (\<lambda>t.-f t)" and "integral (\<lambda>t.-f t) = -integral f"
unfolding integrable_def integral_def by simp_all
qed
lemma integral_times_const:
assumes "integrable f"
shows "integrable (\<lambda>t. a * f t)" (is "?P a")
and "integral (\<lambda>t. a * f t) = a * integral f" (is "?I a")
proof -
{ fix a :: real assume "0 \<le> a"
hence "pos_part (\<lambda>t. a * f t) = (\<lambda>t. a * pos_part f t)"
and "neg_part (\<lambda>t. a * f t) = (\<lambda>t. a * neg_part f t)"
unfolding pos_part_def neg_part_def max_def min_def
by (auto intro!: ext simp: zero_le_mult_iff)
moreover
obtain x y where x: "x \<in> nnfis (pos_part f)" and y: "y \<in> nnfis (neg_part f)"
using assms unfolding integrable_def by auto
ultimately
have "a * x \<in> nnfis (pos_part (\<lambda>t. a * f t))" and
"a * y \<in> nnfis (neg_part (\<lambda>t. a * f t))"
using nnfis_times[OF _ `0 \<le> a`] by auto
with x y have "?P a \<and> ?I a"
unfolding integrable_def integral_def by (auto simp: algebra_simps) }
note int = this
have "?P a \<and> ?I a"
proof (cases "0 \<le> a")
case True from int[OF this] show ?thesis .
next
case False with int[of "- a"] integral_uminus[of "\<lambda>t. - a * f t"]
show ?thesis by auto
qed
thus "integrable (\<lambda>t. a * f t)"
and "integral (\<lambda>t. a * f t) = a * integral f" by simp_all
qed
lemma integral_cmul_indicator:
assumes "s \<in> sets M"
shows "integral (\<lambda>x. c * indicator_fn s x) = c * (measure M s)"
and "integrable (\<lambda>x. c * indicator_fn s x)"
using assms integral_times_const integral_indicator_fn by auto
lemma integral_zero:
shows "integral (\<lambda>x. 0) = 0"
and "integrable (\<lambda>x. 0)"
using integral_cmul_indicator[OF empty_sets, of 0]
unfolding indicator_fn_def by auto
lemma integral_setsum:
assumes "finite S"
assumes "\<And>n. n \<in> S \<Longrightarrow> integrable (f n)"
shows "integral (\<lambda>x. \<Sum> i \<in> S. f i x) = (\<Sum> i \<in> S. integral (f i))" (is "?int S")
and "integrable (\<lambda>x. \<Sum> i \<in> S. f i x)" (is "?I s")
proof -
from assms have "?int S \<and> ?I S"
proof (induct S)
case empty thus ?case by (simp add: integral_zero)
next
case (insert i S)
thus ?case
apply simp
apply (subst integral_add)
using assms apply auto
apply (subst integral_add)
using assms by auto
qed
thus "?int S" and "?I S" by auto
qed
lemma (in measure_space) integrable_abs:
assumes "integrable f"
shows "integrable (\<lambda> x. \<bar>f x\<bar>)"
using assms
proof -
from assms obtain p q where pq: "p \<in> nnfis (pos_part f)" "q \<in> nnfis (neg_part f)"
unfolding integrable_def by auto
hence "p + q \<in> nnfis (\<lambda> x. pos_part f x + neg_part f x)"
using nnfis_add by auto
hence "p + q \<in> nnfis (\<lambda> x. \<bar>f x\<bar>)" using pos_neg_part_abs[of f] by simp
thus ?thesis unfolding integrable_def
using ext[OF pos_part_abs[of f], of "\<lambda> y. y"]
ext[OF neg_part_abs[of f], of "\<lambda> y. y"]
using nnfis_0 by auto
qed
lemma markov_ineq:
assumes "integrable f" "0 < a" "integrable (\<lambda>x. \<bar>f x\<bar>^n)"
shows "measure M (f -` {a ..} \<inter> space M) \<le> integral (\<lambda>x. \<bar>f x\<bar>^n) / a^n"
using assms
proof -
from assms have "0 < a ^ n" using real_root_pow_pos by auto
from assms have "f \<in> borel_measurable M"
using integral_borel_measurable[OF `integrable f`] by auto
hence w: "{w . w \<in> space M \<and> a \<le> f w} \<in> sets M"
using borel_measurable_ge_iff by auto
have i: "integrable (indicator_fn {w . w \<in> space M \<and> a \<le> f w})"
using integral_indicator_fn[OF w] by simp
have v1: "\<And> t. a ^ n * (indicator_fn {w . w \<in> space M \<and> a \<le> f w}) t
\<le> (f t) ^ n * (indicator_fn {w . w \<in> space M \<and> a \<le> f w}) t"
unfolding indicator_fn_def
using `0 < a` power_mono[of a] assms by auto
have v2: "\<And> t. (f t) ^ n * (indicator_fn {w . w \<in> space M \<and> a \<le> f w}) t \<le> \<bar>f t\<bar> ^ n"
unfolding indicator_fn_def
using power_mono[of a _ n] abs_ge_self `a > 0`
by auto
have "{w \<in> space M. a \<le> f w} \<inter> space M = {w . a \<le> f w} \<inter> space M"
using Collect_eq by auto
from Int_absorb2[OF sets_into_space[OF w]] `0 < a ^ n` sets_into_space[OF w] w this
have "(a ^ n) * (measure M ((f -` {y . a \<le> y}) \<inter> space M)) =
(a ^ n) * measure M {w . w \<in> space M \<and> a \<le> f w}"
unfolding vimage_Collect_eq[of f] by simp
also have "\<dots> = integral (\<lambda> t. a ^ n * (indicator_fn {w . w \<in> space M \<and> a \<le> f w}) t)"
using integral_cmul_indicator[OF w] i by auto
also have "\<dots> \<le> integral (\<lambda> t. \<bar> f t \<bar> ^ n)"
apply (rule integral_mono)
using integral_cmul_indicator[OF w]
`integrable (\<lambda> x. \<bar>f x\<bar> ^ n)` real_le_trans[OF v1 v2] by auto
finally show "measure M (f -` {a ..} \<inter> space M) \<le> integral (\<lambda>x. \<bar>f x\<bar>^n) / a^n"
unfolding atLeast_def
by (auto intro!: mult_imp_le_div_pos[OF `0 < a ^ n`], simp add: real_mult_commute)
qed
lemma (in measure_space) integral_0:
fixes f :: "'a \<Rightarrow> real"
assumes "integrable f" "integral f = 0" "nonneg f" and borel: "f \<in> borel_measurable M"
shows "measure M ({x. f x \<noteq> 0} \<inter> space M) = 0"
proof -
have "{x. f x \<noteq> 0} = {x. \<bar>f x\<bar> > 0}" by auto
moreover
{ fix y assume "y \<in> {x. \<bar> f x \<bar> > 0}"
hence "\<bar> f y \<bar> > 0" by auto
hence "\<exists> n. \<bar>f y\<bar> \<ge> inverse (real (Suc n))"
using ex_inverse_of_nat_Suc_less[of "\<bar>f y\<bar>"] less_imp_le unfolding real_of_nat_def by auto
hence "y \<in> (\<Union> n. {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))})"
by auto }
moreover
{ fix y assume "y \<in> (\<Union> n. {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))})"
then obtain n where n: "y \<in> {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))}" by auto
hence "\<bar>f y\<bar> \<ge> inverse (real (Suc n))" by auto
hence "\<bar>f y\<bar> > 0"
using real_of_nat_Suc_gt_zero
positive_imp_inverse_positive[of "real_of_nat (Suc n)"] by fastsimp
hence "y \<in> {x. \<bar>f x\<bar> > 0}" by auto }
ultimately have fneq0_UN: "{x. f x \<noteq> 0} = (\<Union> n. {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))})"
by blast
{ fix n
have int_one: "integrable (\<lambda> x. \<bar>f x\<bar> ^ 1)" using integrable_abs assms by auto
have "measure M (f -` {inverse (real (Suc n))..} \<inter> space M)
\<le> integral (\<lambda> x. \<bar>f x\<bar> ^ 1) / (inverse (real (Suc n)) ^ 1)"
using markov_ineq[OF `integrable f` _ int_one] real_of_nat_Suc_gt_zero by auto
hence le0: "measure M (f -` {inverse (real (Suc n))..} \<inter> space M) \<le> 0"
using assms unfolding nonneg_def by auto
have "{x. f x \<ge> inverse (real (Suc n))} \<inter> space M \<in> sets M"
apply (subst Int_commute) unfolding Int_def
using borel[unfolded borel_measurable_ge_iff] by simp
hence m0: "measure M ({x. f x \<ge> inverse (real (Suc n))} \<inter> space M) = 0 \<and>
{x. f x \<ge> inverse (real (Suc n))} \<inter> space M \<in> sets M"
using positive le0 unfolding atLeast_def by fastsimp }
moreover hence "range (\<lambda> n. {x. f x \<ge> inverse (real (Suc n))} \<inter> space M) \<subseteq> sets M"
by auto
moreover
{ fix n
have "inverse (real (Suc n)) \<ge> inverse (real (Suc (Suc n)))"
using less_imp_inverse_less real_of_nat_Suc_gt_zero[of n] by fastsimp
hence "\<And> x. f x \<ge> inverse (real (Suc n)) \<Longrightarrow> f x \<ge> inverse (real (Suc (Suc n)))" by (rule order_trans)
hence "{x. f x \<ge> inverse (real (Suc n))} \<inter> space M
\<subseteq> {x. f x \<ge> inverse (real (Suc (Suc n)))} \<inter> space M" by auto }
ultimately have "(\<lambda> x. 0) ----> measure M (\<Union> n. {x. f x \<ge> inverse (real (Suc n))} \<inter> space M)"
using monotone_convergence[of "\<lambda> n. {x. f x \<ge> inverse (real (Suc n))} \<inter> space M"]
unfolding o_def by (simp del: of_nat_Suc)
hence "measure M (\<Union> n. {x. f x \<ge> inverse (real (Suc n))} \<inter> space M) = 0"
using LIMSEQ_const[of 0] LIMSEQ_unique by simp
hence "measure M ((\<Union> n. {x. \<bar>f x\<bar> \<ge> inverse (real (Suc n))}) \<inter> space M) = 0"
using assms unfolding nonneg_def by auto
thus "measure M ({x. f x \<noteq> 0} \<inter> space M) = 0" using fneq0_UN by simp
qed
section "Lebesgue integration on countable spaces"
lemma nnfis_on_countable:
assumes borel: "f \<in> borel_measurable M"
and bij: "bij_betw enum S (f ` space M - {0})"
and enum_zero: "enum ` (-S) \<subseteq> {0}"
and nn_enum: "\<And>n. 0 \<le> enum n"
and sums: "(\<lambda>r. enum r * measure M (f -` {enum r} \<inter> space M)) sums x" (is "?sum sums x")
shows "x \<in> nnfis f"
proof -
have inj_enum: "inj_on enum S"
and range_enum: "enum ` S = f ` space M - {0}"
using bij by (auto simp: bij_betw_def)
let "?x n z" = "\<Sum>i = 0..<n. enum i * indicator_fn (f -` {enum i} \<inter> space M) z"
show ?thesis
proof (rule nnfis_mon_conv)
show "(\<lambda>n. \<Sum>i = 0..<n. ?sum i) ----> x" using sums unfolding sums_def .
next
fix n
show "(\<Sum>i = 0..<n. ?sum i) \<in> nnfis (?x n)"
proof (induct n)
case 0 thus ?case by (simp add: nnfis_0)
next
case (Suc n) thus ?case using nn_enum
by (auto intro!: nnfis_add nnfis_times psfis_nnfis[OF psfis_indicator] borel_measurable_vimage[OF borel])
qed
next
show "mono_convergent ?x f (space M)"
proof (rule mono_convergentI)
fix x
show "incseq (\<lambda>n. ?x n x)"
by (rule incseq_SucI, auto simp: indicator_fn_def nn_enum)
have fin: "\<And>n. finite (enum ` ({0..<n} \<inter> S))" by auto
assume "x \<in> space M"
hence "f x \<in> enum ` S \<or> f x = 0" using range_enum by auto
thus "(\<lambda>n. ?x n x) ----> f x"
proof (rule disjE)
assume "f x \<in> enum ` S"
then obtain i where "i \<in> S" and "f x = enum i" by auto
{ fix n
have sum_ranges:
"i < n \<Longrightarrow> enum`({0..<n} \<inter> S) \<inter> {z. enum i = z \<and> x\<in>space M} = {enum i}"
"\<not> i < n \<Longrightarrow> enum`({0..<n} \<inter> S) \<inter> {z. enum i = z \<and> x\<in>space M} = {}"
using `x \<in> space M` `i \<in> S` inj_enum[THEN inj_on_iff] by auto
have "?x n x =
(\<Sum>i \<in> {0..<n} \<inter> S. enum i * indicator_fn (f -` {enum i} \<inter> space M) x)"
using enum_zero by (auto intro!: setsum_mono_zero_cong_right)
also have "... =
(\<Sum>z \<in> enum`({0..<n} \<inter> S). z * indicator_fn (f -` {z} \<inter> space M) x)"
using inj_enum[THEN subset_inj_on] by (auto simp: setsum_reindex)
also have "... = (if i < n then f x else 0)"
unfolding indicator_fn_def if_distrib[where x=1 and y=0]
setsum_cases[OF fin]
using sum_ranges `f x = enum i`
by auto
finally have "?x n x = (if i < n then f x else 0)" . }
note sum_equals_if = this
show ?thesis unfolding sum_equals_if
by (rule LIMSEQ_offset[where k="i + 1"]) (auto intro!: LIMSEQ_const)
next
assume "f x = 0"
{ fix n have "?x n x = 0"
unfolding indicator_fn_def if_distrib[where x=1 and y=0]
setsum_cases[OF finite_atLeastLessThan]
using `f x = 0` `x \<in> space M`
by (auto split: split_if) }
thus ?thesis using `f x = 0` by (auto intro!: LIMSEQ_const)
qed
qed
qed
qed
lemma integral_on_countable:
fixes enum :: "nat \<Rightarrow> real"
assumes borel: "f \<in> borel_measurable M"
and bij: "bij_betw enum S (f ` space M)"
and enum_zero: "enum ` (-S) \<subseteq> {0}"
and abs_summable: "summable (\<lambda>r. \<bar>enum r * measure M (f -` {enum r} \<inter> space M)\<bar>)"
shows "integrable f"
and "integral f = (\<Sum>r. enum r * measure M (f -` {enum r} \<inter> space M))" (is "_ = suminf (?sum f enum)")
proof -
{ fix f enum
assume borel: "f \<in> borel_measurable M"
and bij: "bij_betw enum S (f ` space M)"
and enum_zero: "enum ` (-S) \<subseteq> {0}"
and abs_summable: "summable (\<lambda>r. \<bar>enum r * measure M (f -` {enum r} \<inter> space M)\<bar>)"
have inj_enum: "inj_on enum S" and range_enum: "f ` space M = enum ` S"
using bij unfolding bij_betw_def by auto
have [simp, intro]: "\<And>X. 0 \<le> measure M (f -` {X} \<inter> space M)"
by (rule positive, rule borel_measurable_vimage[OF borel])
have "(\<Sum>r. ?sum (pos_part f) (pos_part enum) r) \<in> nnfis (pos_part f) \<and>
summable (\<lambda>r. ?sum (pos_part f) (pos_part enum) r)"
proof (rule conjI, rule nnfis_on_countable)
have pos_f_image: "pos_part f ` space M - {0} = f ` space M \<inter> {0<..}"
unfolding pos_part_def max_def by auto
show "bij_betw (pos_part enum) {x \<in> S. 0 < enum x} (pos_part f ` space M - {0})"
unfolding bij_betw_def pos_f_image
unfolding pos_part_def max_def range_enum
by (auto intro!: inj_onI simp: inj_enum[THEN inj_on_eq_iff])
show "\<And>n. 0 \<le> pos_part enum n" unfolding pos_part_def max_def by auto
show "pos_part f \<in> borel_measurable M"
by (rule pos_part_borel_measurable[OF borel])
show "pos_part enum ` (- {x \<in> S. 0 < enum x}) \<subseteq> {0}"
unfolding pos_part_def max_def using enum_zero by auto
show "summable (\<lambda>r. ?sum (pos_part f) (pos_part enum) r)"
proof (rule summable_comparison_test[OF _ abs_summable], safe intro!: exI[of _ 0])
fix n :: nat
have "pos_part enum n \<noteq> 0 \<Longrightarrow> (pos_part f -` {enum n} \<inter> space M) =
(if 0 < enum n then (f -` {enum n} \<inter> space M) else {})"
unfolding pos_part_def max_def by (auto split: split_if_asm)
thus "norm (?sum (pos_part f) (pos_part enum) n) \<le> \<bar>?sum f enum n \<bar>"
by (cases "pos_part enum n = 0",
auto simp: pos_part_def max_def abs_mult not_le split: split_if_asm intro!: mult_nonpos_nonneg)
qed
thus "(\<lambda>r. ?sum (pos_part f) (pos_part enum) r) sums (\<Sum>r. ?sum (pos_part f) (pos_part enum) r)"
by (rule summable_sums)
qed }
note pos = this
note pos_part = pos[OF assms(1-4)]
moreover
have neg_part_to_pos_part:
"\<And>f :: _ \<Rightarrow> real. neg_part f = pos_part (uminus \<circ> f)"
by (auto simp: pos_part_def neg_part_def min_def max_def expand_fun_eq)
have neg_part: "(\<Sum>r. ?sum (neg_part f) (neg_part enum) r) \<in> nnfis (neg_part f) \<and>
summable (\<lambda>r. ?sum (neg_part f) (neg_part enum) r)"
unfolding neg_part_to_pos_part
proof (rule pos)
show "uminus \<circ> f \<in> borel_measurable M" unfolding comp_def
by (rule borel_measurable_uminus_borel_measurable[OF borel])
show "bij_betw (uminus \<circ> enum) S ((uminus \<circ> f) ` space M)"
using bij unfolding bij_betw_def
by (auto intro!: comp_inj_on simp: image_compose)
show "(uminus \<circ> enum) ` (- S) \<subseteq> {0}"
using enum_zero by auto
have minus_image: "\<And>r. (uminus \<circ> f) -` {(uminus \<circ> enum) r} \<inter> space M = f -` {enum r} \<inter> space M"
by auto
show "summable (\<lambda>r. \<bar>(uminus \<circ> enum) r * measure_space.measure M ((uminus \<circ> f) -` {(uminus \<circ> enum) r} \<inter> space M)\<bar>)"
unfolding minus_image using abs_summable by simp
qed
ultimately show "integrable f" unfolding integrable_def by auto
{ fix r
have "?sum (pos_part f) (pos_part enum) r - ?sum (neg_part f) (neg_part enum) r = ?sum f enum r"
proof (cases rule: linorder_cases)
assume "0 < enum r"
hence "pos_part f -` {enum r} \<inter> space M = f -` {enum r} \<inter> space M"
unfolding pos_part_def max_def by (auto split: split_if_asm)
with `0 < enum r` show ?thesis unfolding pos_part_def neg_part_def min_def max_def expand_fun_eq
by auto
next
assume "enum r < 0"
hence "neg_part f -` {- enum r} \<inter> space M = f -` {enum r} \<inter> space M"
unfolding neg_part_def min_def by (auto split: split_if_asm)
with `enum r < 0` show ?thesis unfolding pos_part_def neg_part_def min_def max_def expand_fun_eq
by auto
qed (simp add: neg_part_def pos_part_def) }
note sum_diff_eq_sum = this
have "(\<Sum>r. ?sum (pos_part f) (pos_part enum) r) - (\<Sum>r. ?sum (neg_part f) (neg_part enum) r)
= (\<Sum>r. ?sum (pos_part f) (pos_part enum) r - ?sum (neg_part f) (neg_part enum) r)"
using neg_part pos_part by (auto intro: suminf_diff)
also have "... = (\<Sum>r. ?sum f enum r)" unfolding sum_diff_eq_sum ..
finally show "integral f = suminf (?sum f enum)"
unfolding integral_def using pos_part neg_part
by (auto dest: the_nnfis)
qed
section "Lebesgue integration on finite space"
lemma integral_finite_on_sets:
assumes "f \<in> borel_measurable M" and "finite (space M)" and "a \<in> sets M"
shows "integral (\<lambda>x. f x * indicator_fn a x) =
(\<Sum> r \<in> f`a. r * measure M (f -` {r} \<inter> a))" (is "integral ?f = _")
proof -
{ fix x assume "x \<in> a"
with assms have "f -` {f x} \<inter> space M \<in> sets M"
by (subst Int_commute)
(auto simp: vimage_def Int_def
intro!: borel_measurable_const
borel_measurable_eq_borel_measurable)
from Int[OF this assms(3)]
sets_into_space[OF assms(3), THEN Int_absorb1]
have "f -` {f x} \<inter> a \<in> sets M" by (simp add: Int_assoc) }
note vimage_f = this
have "finite a"
using assms(2,3) sets_into_space
by (auto intro: finite_subset)
have "integral (\<lambda>x. f x * indicator_fn a x) =
integral (\<lambda>x. \<Sum>i\<in>f ` a. i * indicator_fn (f -` {i} \<inter> a) x)"
(is "_ = integral (\<lambda>x. setsum (?f x) _)")
unfolding indicator_fn_def if_distrib
using `finite a` by (auto simp: setsum_cases intro!: integral_cong)
also have "\<dots> = (\<Sum>i\<in>f`a. integral (\<lambda>x. ?f x i))"
proof (rule integral_setsum, safe)
fix n x assume "x \<in> a"
thus "integrable (\<lambda>y. ?f y (f x))"
using integral_indicator_fn(2)[OF vimage_f]
by (auto intro!: integral_times_const)
qed (simp add: `finite a`)
also have "\<dots> = (\<Sum>i\<in>f`a. i * measure M (f -` {i} \<inter> a))"
using integral_cmul_indicator[OF vimage_f]
by (auto intro!: setsum_cong)
finally show ?thesis .
qed
lemma integral_finite:
assumes "f \<in> borel_measurable M" and "finite (space M)"
shows "integral f = (\<Sum> r \<in> f ` space M. r * measure M (f -` {r} \<inter> space M))"
using integral_finite_on_sets[OF assms top]
integral_cong[of "\<lambda>x. f x * indicator_fn (space M) x" f]
by (auto simp add: indicator_fn_def)
section "Radon–Nikodym derivative"
definition
"RN_deriv v \<equiv> SOME f. measure_space (M\<lparr>measure := v\<rparr>) \<and>
f \<in> borel_measurable M \<and>
(\<forall>a \<in> sets M. (integral (\<lambda>x. f x * indicator_fn a x) = v a))"
end
lemma sigma_algebra_cong:
fixes M :: "('a, 'b) algebra_scheme" and M' :: "('a, 'c) algebra_scheme"
assumes *: "sigma_algebra M"
and cong: "space M = space M'" "sets M = sets M'"
shows "sigma_algebra M'"
using * unfolding sigma_algebra_def algebra_def sigma_algebra_axioms_def unfolding cong .
lemma finite_Pow_additivity_sufficient:
assumes "finite (space M)" and "sets M = Pow (space M)"
and "positive M (measure M)" and "additive M (measure M)"
shows "finite_measure_space M"
proof -
have "sigma_algebra M"
using assms by (auto intro!: sigma_algebra_cong[OF sigma_algebra_Pow])
have "measure_space M"
by (rule Measure.finite_additivity_sufficient) (fact+)
thus ?thesis
unfolding finite_measure_space_def finite_measure_space_axioms_def
using assms by simp
qed
lemma finite_measure_spaceI:
assumes "measure_space M" and "finite (space M)" and "sets M = Pow (space M)"
shows "finite_measure_space M"
unfolding finite_measure_space_def finite_measure_space_axioms_def
using assms by simp
lemma (in finite_measure_space) integral_finite_singleton:
"integral f = (\<Sum>x \<in> space M. f x * measure M {x})"
proof -
have "f \<in> borel_measurable M"
unfolding borel_measurable_le_iff
using sets_eq_Pow by auto
{ fix r let ?x = "f -` {r} \<inter> space M"
have "?x \<subseteq> space M" by auto
with finite_space sets_eq_Pow have "measure M ?x = (\<Sum>i \<in> ?x. measure M {i})"
by (auto intro!: measure_real_sum_image) }
note measure_eq_setsum = this
show ?thesis
unfolding integral_finite[OF `f \<in> borel_measurable M` finite_space]
measure_eq_setsum setsum_right_distrib
apply (subst setsum_Sigma)
apply (simp add: finite_space)
apply (simp add: finite_space)
proof (rule setsum_reindex_cong[symmetric])
fix a assume "a \<in> Sigma (f ` space M) (\<lambda>x. f -` {x} \<inter> space M)"
thus "(\<lambda>(x, y). x * measure M {y}) a = f (snd a) * measure_space.measure M {snd a}"
by auto
qed (auto intro!: image_eqI inj_onI)
qed
lemma (in finite_measure_space) RN_deriv_finite_singleton:
fixes v :: "'a set \<Rightarrow> real"
assumes ms_v: "measure_space (M\<lparr>measure := v\<rparr>)"
and eq_0: "\<And>x. \<lbrakk> x \<in> space M ; measure M {x} = 0 \<rbrakk> \<Longrightarrow> v {x} = 0"
and "x \<in> space M" and "measure M {x} \<noteq> 0"
shows "RN_deriv v x = v {x} / (measure M {x})" (is "_ = ?v x")
unfolding RN_deriv_def
proof (rule someI2_ex[where Q = "\<lambda>f. f x = ?v x"], rule exI[where x = ?v], safe)
show "(\<lambda>a. v {a} / measure_space.measure M {a}) \<in> borel_measurable M"
unfolding borel_measurable_le_iff using sets_eq_Pow by auto
next
fix a assume "a \<in> sets M"
hence "a \<subseteq> space M" and "finite a"
using sets_into_space finite_space by (auto intro: finite_subset)
have *: "\<And>x a. x \<in> space M \<Longrightarrow> (if measure M {x} = 0 then 0 else v {x} * indicator_fn a x) =
v {x} * indicator_fn a x" using eq_0 by auto
from measure_space.measure_real_sum_image[OF ms_v, of a]
sets_eq_Pow `a \<in> sets M` sets_into_space `finite a`
have "v a = (\<Sum>x\<in>a. v {x})" by auto
thus "integral (\<lambda>x. v {x} / measure_space.measure M {x} * indicator_fn a x) = v a"
apply (simp add: eq_0 integral_finite_singleton)
apply (unfold divide_1)
by (simp add: * indicator_fn_def if_distrib setsum_cases finite_space `a \<subseteq> space M` Int_absorb1)
next
fix w assume "w \<in> borel_measurable M"
assume int_eq_v: "\<forall>a\<in>sets M. integral (\<lambda>x. w x * indicator_fn a x) = v a"
have "{x} \<in> sets M" using sets_eq_Pow `x \<in> space M` by auto
have "w x * measure M {x} =
(\<Sum>y\<in>space M. w y * indicator_fn {x} y * measure M {y})"
apply (subst (3) mult_commute)
unfolding indicator_fn_def if_distrib setsum_cases[OF finite_space]
using `x \<in> space M` by simp
also have "... = v {x}"
using int_eq_v[rule_format, OF `{x} \<in> sets M`]
by (simp add: integral_finite_singleton)
finally show "w x = v {x} / measure M {x}"
using `measure M {x} \<noteq> 0` by (simp add: eq_divide_eq)
qed fact
end