Metamath Proof Explorer

Theorem meaiuninc

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 meaiuninc.m ⊢ φ → M ∈ Meas
meaiuninc.n ⊢ φ → N ∈ ℤ
meaiuninc.z ⊢ Z = ℤ ≥ N
meaiuninc.e ⊢ φ → E : Z ⟶ dom ⁡ M
meaiuninc.i ⊢ φ ∧ n ∈ Z → E ⁡ n ⊆ E ⁡ n + 1
meaiuninc.x ⊢ φ → ∃ x ∈ ℝ ∀ n ∈ Z M ⁡ E ⁡ n ≤ x
meaiuninc.s ⊢ S = n ∈ Z ⟼ M ⁡ E ⁡ n
Assertion meaiuninc ⊢ φ → S ⇝ M ⁡ ⋃ n ∈ Z E ⁡ n

Proof

Step Hyp Ref Expression
1 meaiuninc.m ⊢ φ → M ∈ Meas
2 meaiuninc.n ⊢ φ → N ∈ ℤ
3 meaiuninc.z ⊢ Z = ℤ ≥ N
4 meaiuninc.e ⊢ φ → E : Z ⟶ dom ⁡ M
5 meaiuninc.i ⊢ φ ∧ n ∈ Z → E ⁡ n ⊆ E ⁡ n + 1
6 meaiuninc.x ⊢ φ → ∃ x ∈ ℝ ∀ n ∈ Z M ⁡ E ⁡ n ≤ x
7 meaiuninc.s ⊢ S = n ∈ Z ⟼ M ⁡ E ⁡ n
8 2fveq3 ⊢ n = m → M ⁡ E ⁡ n = M ⁡ E ⁡ m
9 8 cbvmptv ⊢ n ∈ Z ⟼ M ⁡ E ⁡ n = m ∈ Z ⟼ M ⁡ E ⁡ m
10 7 9 eqtri ⊢ S = m ∈ Z ⟼ M ⁡ E ⁡ m
11 10 a1i ⊢ φ → S = m ∈ Z ⟼ M ⁡ E ⁡ m
12 10 7 eqtr3i ⊢ m ∈ Z ⟼ M ⁡ E ⁡ m = n ∈ Z ⟼ M ⁡ E ⁡ n
13 fveq2 ⊢ k = i → E ⁡ k = E ⁡ i
14 13 cbviunv ⊢ ⋃ k ∈ N ..^ m E ⁡ k = ⋃ i ∈ N ..^ m E ⁡ i
15 14 difeq2i ⊢ E ⁡ m ∖ ⋃ k ∈ N ..^ m E ⁡ k = E ⁡ m ∖ ⋃ i ∈ N ..^ m E ⁡ i
16 15 mpteq2i ⊢ m ∈ Z ⟼ E ⁡ m ∖ ⋃ k ∈ N ..^ m E ⁡ k = m ∈ Z ⟼ E ⁡ m ∖ ⋃ i ∈ N ..^ m E ⁡ i
17 fveq2 ⊢ m = n → E ⁡ m = E ⁡ n
18 oveq2 ⊢ m = n → N ..^ m = N ..^ n
19 18 iuneq1d ⊢ m = n → ⋃ i ∈ N ..^ m E ⁡ i = ⋃ i ∈ N ..^ n E ⁡ i
20 17 19 difeq12d ⊢ m = n → E ⁡ m ∖ ⋃ i ∈ N ..^ m E ⁡ i = E ⁡ n ∖ ⋃ i ∈ N ..^ n E ⁡ i
21 20 cbvmptv ⊢ m ∈ Z ⟼ E ⁡ m ∖ ⋃ i ∈ N ..^ m E ⁡ i = n ∈ Z ⟼ E ⁡ n ∖ ⋃ i ∈ N ..^ n E ⁡ i
22 16 21 eqtri ⊢ m ∈ Z ⟼ E ⁡ m ∖ ⋃ k ∈ N ..^ m E ⁡ k = n ∈ Z ⟼ E ⁡ n ∖ ⋃ i ∈ N ..^ n E ⁡ i
23 1 2 3 4 5 6 12 22 meaiuninclem ⊢ φ → m ∈ Z ⟼ M ⁡ E ⁡ m ⇝ M ⁡ ⋃ n ∈ Z E ⁡ n
24 11 23 eqbrtrd ⊢ φ → S ⇝ M ⁡ ⋃ n ∈ Z E ⁡ n