header {* Sigma algebras *}
theory Sigma_Algebra
imports Main
begin
text {*
This is just a tiny example demonstrating the use of inductive
definitions in classical mathematics. We define the least @{text
σ}-algebra over a given set of sets.
*}
inductive_set σ_algebra :: "'a set set => 'a set set" for A :: "'a set set"
where
basic: "a ∈ A ==> a ∈ σ_algebra A"
| UNIV: "UNIV ∈ σ_algebra A"
| complement: "a ∈ σ_algebra A ==> -a ∈ σ_algebra A"
| Union: "(!!i::nat. a i ∈ σ_algebra A) ==> (\<Union>i. a i) ∈ σ_algebra A"
text {*
The following basic facts are consequences of the closure properties
of any @{text σ}-algebra, merely using the introduction rules, but
no induction nor cases.
*}
theorem sigma_algebra_empty: "{} ∈ σ_algebra A"
proof -
have "UNIV ∈ σ_algebra A" by (rule σ_algebra.UNIV)
then have "-UNIV ∈ σ_algebra A" by (rule σ_algebra.complement)
also have "-UNIV = {}" by simp
finally show ?thesis .
qed
theorem sigma_algebra_Inter:
"(!!i::nat. a i ∈ σ_algebra A) ==> (\<Inter>i. a i) ∈ σ_algebra A"
proof -
assume "!!i::nat. a i ∈ σ_algebra A"
then have "!!i::nat. -(a i) ∈ σ_algebra A" by (rule σ_algebra.complement)
then have "(\<Union>i. -(a i)) ∈ σ_algebra A" by (rule σ_algebra.Union)
then have "-(\<Union>i. -(a i)) ∈ σ_algebra A" by (rule σ_algebra.complement)
also have "-(\<Union>i. -(a i)) = (\<Inter>i. a i)" by simp
finally show ?thesis .
qed
end