Metamath Proof Explorer


Theorem meaiininc2

Description: Measures are continuous from above: if E is a nonincreasing sequence of measurable sets, and any of the sets has finite measure, then the measure of the intersection is the limit of the measures. This is Proposition 112C (f) of Fremlin1 p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021)

Ref Expression
Hypotheses meaiininc2.f 𝑛 𝜑
meaiininc2.p 𝑘 𝜑
meaiininc2.m ( 𝜑𝑀 ∈ Meas )
meaiininc2.n ( 𝜑𝑁 ∈ ℤ )
meaiininc2.z 𝑍 = ( ℤ𝑁 )
meaiininc2.e ( 𝜑𝐸 : 𝑍 ⟶ dom 𝑀 )
meaiininc2.i ( ( 𝜑𝑛𝑍 ) → ( 𝐸 ‘ ( 𝑛 + 1 ) ) ⊆ ( 𝐸𝑛 ) )
meaiininc2.k ( 𝜑 → ∃ 𝑘𝑍 ( 𝑀 ‘ ( 𝐸𝑘 ) ) ∈ ℝ )
meaiininc2.s 𝑆 = ( 𝑛𝑍 ↦ ( 𝑀 ‘ ( 𝐸𝑛 ) ) )
Assertion meaiininc2 ( 𝜑𝑆 ⇝ ( 𝑀 𝑛𝑍 ( 𝐸𝑛 ) ) )

Proof

Step Hyp Ref Expression
1 meaiininc2.f 𝑛 𝜑
2 meaiininc2.p 𝑘 𝜑
3 meaiininc2.m ( 𝜑𝑀 ∈ Meas )
4 meaiininc2.n ( 𝜑𝑁 ∈ ℤ )
5 meaiininc2.z 𝑍 = ( ℤ𝑁 )
6 meaiininc2.e ( 𝜑𝐸 : 𝑍 ⟶ dom 𝑀 )
7 meaiininc2.i ( ( 𝜑𝑛𝑍 ) → ( 𝐸 ‘ ( 𝑛 + 1 ) ) ⊆ ( 𝐸𝑛 ) )
8 meaiininc2.k ( 𝜑 → ∃ 𝑘𝑍 ( 𝑀 ‘ ( 𝐸𝑘 ) ) ∈ ℝ )
9 meaiininc2.s 𝑆 = ( 𝑛𝑍 ↦ ( 𝑀 ‘ ( 𝐸𝑛 ) ) )
10 nfv 𝑘 𝑆 ⇝ ( 𝑀 𝑛𝑍 ( 𝐸𝑛 ) )
11 nfv 𝑛 𝑘𝑍
12 nfv 𝑛 ( 𝑀 ‘ ( 𝐸𝑘 ) ) ∈ ℝ
13 1 11 12 nf3an 𝑛 ( 𝜑𝑘𝑍 ∧ ( 𝑀 ‘ ( 𝐸𝑘 ) ) ∈ ℝ )
14 3 3ad2ant1 ( ( 𝜑𝑘𝑍 ∧ ( 𝑀 ‘ ( 𝐸𝑘 ) ) ∈ ℝ ) → 𝑀 ∈ Meas )
15 4 3ad2ant1 ( ( 𝜑𝑘𝑍 ∧ ( 𝑀 ‘ ( 𝐸𝑘 ) ) ∈ ℝ ) → 𝑁 ∈ ℤ )
16 6 3ad2ant1 ( ( 𝜑𝑘𝑍 ∧ ( 𝑀 ‘ ( 𝐸𝑘 ) ) ∈ ℝ ) → 𝐸 : 𝑍 ⟶ dom 𝑀 )
17 7 3ad2antl1 ( ( ( 𝜑𝑘𝑍 ∧ ( 𝑀 ‘ ( 𝐸𝑘 ) ) ∈ ℝ ) ∧ 𝑛𝑍 ) → ( 𝐸 ‘ ( 𝑛 + 1 ) ) ⊆ ( 𝐸𝑛 ) )
18 id ( 𝑘𝑍𝑘𝑍 )
19 18 5 eleqtrdi ( 𝑘𝑍𝑘 ∈ ( ℤ𝑁 ) )
20 19 3ad2ant2 ( ( 𝜑𝑘𝑍 ∧ ( 𝑀 ‘ ( 𝐸𝑘 ) ) ∈ ℝ ) → 𝑘 ∈ ( ℤ𝑁 ) )
21 simp3 ( ( 𝜑𝑘𝑍 ∧ ( 𝑀 ‘ ( 𝐸𝑘 ) ) ∈ ℝ ) → ( 𝑀 ‘ ( 𝐸𝑘 ) ) ∈ ℝ )
22 13 14 15 5 16 17 20 21 9 meaiininc ( ( 𝜑𝑘𝑍 ∧ ( 𝑀 ‘ ( 𝐸𝑘 ) ) ∈ ℝ ) → 𝑆 ⇝ ( 𝑀 𝑛𝑍 ( 𝐸𝑛 ) ) )
23 22 3exp ( 𝜑 → ( 𝑘𝑍 → ( ( 𝑀 ‘ ( 𝐸𝑘 ) ) ∈ ℝ → 𝑆 ⇝ ( 𝑀 𝑛𝑍 ( 𝐸𝑛 ) ) ) ) )
24 2 10 23 rexlimd ( 𝜑 → ( ∃ 𝑘𝑍 ( 𝑀 ‘ ( 𝐸𝑘 ) ) ∈ ℝ → 𝑆 ⇝ ( 𝑀 𝑛𝑍 ( 𝐸𝑛 ) ) ) )
25 8 24 mpd ( 𝜑𝑆 ⇝ ( 𝑀 𝑛𝑍 ( 𝐸𝑛 ) ) )