HOL-Analysis: add measurable sets with finite measures, prove affine transformation rule for the Lebesgue measure
(* Title: HOL/Library/Indicator_Function.thy
Author: Johannes Hoelzl (TU Muenchen)
*)
section \<open>Indicator Function\<close>
theory Indicator_Function
imports Complex_Main Disjoint_Sets
begin
definition "indicator S x = (if x \<in> S then 1 else 0)"
lemma indicator_simps[simp]:
"x \<in> S \<Longrightarrow> indicator S x = 1"
"x \<notin> S \<Longrightarrow> indicator S x = 0"
unfolding indicator_def by auto
lemma indicator_pos_le[intro, simp]: "(0::'a::linordered_semidom) \<le> indicator S x"
and indicator_le_1[intro, simp]: "indicator S x \<le> (1::'a::linordered_semidom)"
unfolding indicator_def by auto
lemma indicator_abs_le_1: "\<bar>indicator S x\<bar> \<le> (1::'a::linordered_idom)"
unfolding indicator_def by auto
lemma indicator_eq_0_iff: "indicator A x = (0::'a::zero_neq_one) \<longleftrightarrow> x \<notin> A"
by (auto simp: indicator_def)
lemma indicator_eq_1_iff: "indicator A x = (1::'a::zero_neq_one) \<longleftrightarrow> x \<in> A"
by (auto simp: indicator_def)
lemma indicator_UNIV [simp]: "indicator UNIV = (\<lambda>x. 1)"
by auto
lemma indicator_leI:
"(x \<in> A \<Longrightarrow> y \<in> B) \<Longrightarrow> (indicator A x :: 'a::linordered_nonzero_semiring) \<le> indicator B y"
by (auto simp: indicator_def)
lemma split_indicator: "P (indicator S x) \<longleftrightarrow> ((x \<in> S \<longrightarrow> P 1) \<and> (x \<notin> S \<longrightarrow> P 0))"
unfolding indicator_def by auto
lemma split_indicator_asm: "P (indicator S x) \<longleftrightarrow> (\<not> (x \<in> S \<and> \<not> P 1 \<or> x \<notin> S \<and> \<not> P 0))"
unfolding indicator_def by auto
lemma indicator_inter_arith: "indicator (A \<inter> B) x = indicator A x * (indicator B x::'a::semiring_1)"
unfolding indicator_def by (auto simp: min_def max_def)
lemma indicator_union_arith:
"indicator (A \<union> B) x = indicator A x + indicator B x - indicator A x * (indicator B x :: 'a::ring_1)"
unfolding indicator_def by (auto simp: min_def max_def)
lemma indicator_inter_min: "indicator (A \<inter> B) x = min (indicator A x) (indicator B x::'a::linordered_semidom)"
and indicator_union_max: "indicator (A \<union> B) x = max (indicator A x) (indicator B x::'a::linordered_semidom)"
unfolding indicator_def by (auto simp: min_def max_def)
lemma indicator_disj_union:
"A \<inter> B = {} \<Longrightarrow> indicator (A \<union> B) x = (indicator A x + indicator B x :: 'a::linordered_semidom)"
by (auto split: split_indicator)
lemma indicator_compl: "indicator (- A) x = 1 - (indicator A x :: 'a::ring_1)"
and indicator_diff: "indicator (A - B) x = indicator A x * (1 - indicator B x ::'a::ring_1)"
unfolding indicator_def by (auto simp: min_def max_def)
lemma indicator_times:
"indicator (A \<times> B) x = indicator A (fst x) * (indicator B (snd x) :: 'a::semiring_1)"
unfolding indicator_def by (cases x) auto
lemma indicator_sum:
"indicator (A <+> B) x = (case x of Inl x \<Rightarrow> indicator A x | Inr x \<Rightarrow> indicator B x)"
unfolding indicator_def by (cases x) auto
lemma indicator_image: "inj f \<Longrightarrow> indicator (f ` X) (f x) = (indicator X x::_::zero_neq_one)"
by (auto simp: indicator_def inj_on_def)
lemma indicator_vimage: "indicator (f -` A) x = indicator A (f x)"
by (auto split: split_indicator)
lemma (* FIXME unnamed!? *)
fixes f :: "'a \<Rightarrow> 'b::semiring_1"
assumes "finite A"
shows setsum_mult_indicator[simp]: "(\<Sum>x \<in> A. f x * indicator B x) = (\<Sum>x \<in> A \<inter> B. f x)"
and setsum_indicator_mult[simp]: "(\<Sum>x \<in> A. indicator B x * f x) = (\<Sum>x \<in> A \<inter> B. f x)"
unfolding indicator_def
using assms by (auto intro!: setsum.mono_neutral_cong_right split: if_split_asm)
lemma setsum_indicator_eq_card:
assumes "finite A"
shows "(\<Sum>x \<in> A. indicator B x) = card (A Int B)"
using setsum_mult_indicator [OF assms, of "\<lambda>x. 1::nat"]
unfolding card_eq_setsum by simp
lemma setsum_indicator_scaleR[simp]:
"finite A \<Longrightarrow>
(\<Sum>x \<in> A. indicator (B x) (g x) *\<^sub>R f x) = (\<Sum>x \<in> {x\<in>A. g x \<in> B x}. f x :: 'a::real_vector)"
by (auto intro!: setsum.mono_neutral_cong_right split: if_split_asm simp: indicator_def)
lemma LIMSEQ_indicator_incseq:
assumes "incseq A"
shows "(\<lambda>i. indicator (A i) x :: 'a::{topological_space,one,zero}) \<longlonglongrightarrow> indicator (\<Union>i. A i) x"
proof (cases "\<exists>i. x \<in> A i")
case True
then obtain i where "x \<in> A i"
by auto
then have *:
"\<And>n. (indicator (A (n + i)) x :: 'a) = 1"
"(indicator (\<Union>i. A i) x :: 'a) = 1"
using incseqD[OF \<open>incseq A\<close>, of i "n + i" for n] \<open>x \<in> A i\<close> by (auto simp: indicator_def)
show ?thesis
by (rule LIMSEQ_offset[of _ i]) (use * in simp)
next
case False
then show ?thesis by (simp add: indicator_def)
qed
lemma LIMSEQ_indicator_UN:
"(\<lambda>k. indicator (\<Union>i<k. A i) x :: 'a::{topological_space,one,zero}) \<longlonglongrightarrow> indicator (\<Union>i. A i) x"
proof -
have "(\<lambda>k. indicator (\<Union>i<k. A i) x::'a) \<longlonglongrightarrow> indicator (\<Union>k. \<Union>i<k. A i) x"
by (intro LIMSEQ_indicator_incseq) (auto simp: incseq_def intro: less_le_trans)
also have "(\<Union>k. \<Union>i<k. A i) = (\<Union>i. A i)"
by auto
finally show ?thesis .
qed
lemma LIMSEQ_indicator_decseq:
assumes "decseq A"
shows "(\<lambda>i. indicator (A i) x :: 'a::{topological_space,one,zero}) \<longlonglongrightarrow> indicator (\<Inter>i. A i) x"
proof (cases "\<exists>i. x \<notin> A i")
case True
then obtain i where "x \<notin> A i"
by auto
then have *:
"\<And>n. (indicator (A (n + i)) x :: 'a) = 0"
"(indicator (\<Inter>i. A i) x :: 'a) = 0"
using decseqD[OF \<open>decseq A\<close>, of i "n + i" for n] \<open>x \<notin> A i\<close> by (auto simp: indicator_def)
show ?thesis
by (rule LIMSEQ_offset[of _ i]) (use * in simp)
next
case False
then show ?thesis by (simp add: indicator_def)
qed
lemma LIMSEQ_indicator_INT:
"(\<lambda>k. indicator (\<Inter>i<k. A i) x :: 'a::{topological_space,one,zero}) \<longlonglongrightarrow> indicator (\<Inter>i. A i) x"
proof -
have "(\<lambda>k. indicator (\<Inter>i<k. A i) x::'a) \<longlonglongrightarrow> indicator (\<Inter>k. \<Inter>i<k. A i) x"
by (intro LIMSEQ_indicator_decseq) (auto simp: decseq_def intro: less_le_trans)
also have "(\<Inter>k. \<Inter>i<k. A i) = (\<Inter>i. A i)"
by auto
finally show ?thesis .
qed
lemma indicator_add:
"A \<inter> B = {} \<Longrightarrow> (indicator A x::_::monoid_add) + indicator B x = indicator (A \<union> B) x"
unfolding indicator_def by auto
lemma of_real_indicator: "of_real (indicator A x) = indicator A x"
by (simp split: split_indicator)
lemma real_of_nat_indicator: "real (indicator A x :: nat) = indicator A x"
by (simp split: split_indicator)
lemma abs_indicator: "\<bar>indicator A x :: 'a::linordered_idom\<bar> = indicator A x"
by (simp split: split_indicator)
lemma mult_indicator_subset:
"A \<subseteq> B \<Longrightarrow> indicator A x * indicator B x = (indicator A x :: 'a::comm_semiring_1)"
by (auto split: split_indicator simp: fun_eq_iff)
lemma indicator_sums:
assumes "\<And>i j. i \<noteq> j \<Longrightarrow> A i \<inter> A j = {}"
shows "(\<lambda>i. indicator (A i) x::real) sums indicator (\<Union>i. A i) x"
proof (cases "\<exists>i. x \<in> A i")
case True
then obtain i where i: "x \<in> A i" ..
with assms have "(\<lambda>i. indicator (A i) x::real) sums (\<Sum>i\<in>{i}. indicator (A i) x)"
by (intro sums_finite) (auto split: split_indicator)
also have "(\<Sum>i\<in>{i}. indicator (A i) x) = indicator (\<Union>i. A i) x"
using i by (auto split: split_indicator)
finally show ?thesis .
next
case False
then show ?thesis by simp
qed
text \<open>
The indicator function of the union of a disjoint family of sets is the
sum over all the individual indicators.
\<close>
lemma indicator_UN_disjoint:
"finite A \<Longrightarrow> disjoint_family_on f A \<Longrightarrow> indicator (UNION A f) x = (\<Sum>y\<in>A. indicator (f y) x)"
by (induct A rule: finite_induct)
(auto simp: disjoint_family_on_def indicator_def split: if_splits)
end