Metamath Proof Explorer

Theorem meaiuninc2

Description: Measures are continuous from below (bounded case): if E is a sequence of nondecreasing measurable sets (with bounded measure) then the measure of the union is the limit of the measures. This is Proposition 112C (e) of Fremlin1 p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021)

Ref Expression
Hypotheses meaiuninc2.m ⊢ φ → M ∈ Meas
meaiuninc2.n ⊢ φ → N ∈ ℤ
meaiuninc2.z ⊢ Z = ℤ ≥ N
meaiuninc2.e ⊢ φ → E : Z ⟶ dom ⁡ M
meaiuninc2.i ⊢ φ ∧ n ∈ Z → E ⁡ n ⊆ E ⁡ n + 1
meaiuninc2.b ⊢ φ → B ∈ ℝ
meaiuninc2.x ⊢ φ ∧ n ∈ Z → M ⁡ E ⁡ n ≤ B
meaiuninc2.s ⊢ S = n ∈ Z ⟼ M ⁡ E ⁡ n
Assertion meaiuninc2 ⊢ φ → S ⇝ M ⁡ ⋃ n ∈ Z E ⁡ n

Proof

Step Hyp Ref Expression
1 meaiuninc2.m ⊢ φ → M ∈ Meas
2 meaiuninc2.n ⊢ φ → N ∈ ℤ
3 meaiuninc2.z ⊢ Z = ℤ ≥ N
4 meaiuninc2.e ⊢ φ → E : Z ⟶ dom ⁡ M
5 meaiuninc2.i ⊢ φ ∧ n ∈ Z → E ⁡ n ⊆ E ⁡ n + 1
6 meaiuninc2.b ⊢ φ → B ∈ ℝ
7 meaiuninc2.x ⊢ φ ∧ n ∈ Z → M ⁡ E ⁡ n ≤ B
8 meaiuninc2.s ⊢ S = n ∈ Z ⟼ M ⁡ E ⁡ n
9 7 ralrimiva ⊢ φ → ∀ n ∈ Z M ⁡ E ⁡ n ≤ B
10 brralrspcev ⊢ B ∈ ℝ ∧ ∀ n ∈ Z M ⁡ E ⁡ n ≤ B → ∃ x ∈ ℝ ∀ n ∈ Z M ⁡ E ⁡ n ≤ x
11 6 9 10 syl2anc ⊢ φ → ∃ x ∈ ℝ ∀ n ∈ Z M ⁡ E ⁡ n ≤ x
12 1 2 3 4 5 11 8 meaiuninc ⊢ φ → S ⇝ M ⁡ ⋃ n ∈ Z E ⁡ n