Let (X, F, µ) be a measurable space, where X is a set, F is a sigma-algebra on X, and μ is a measure on F. Let A₁, A2, ..., An be a finite sequence of pairwise disjoint sets in F with corresponding measures µ(A1₁), µ(A2), …‚µ(An). Prove that for any measurable set EC X, the following holds: µ (E^Ư²₁²_₁ A₁) = Σ²²_₁µ(EÑA₂)

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Let (X, F, μ) be a measurable space, where X is a set, F is a sigma-algebra on X, and μ
is a measure on F. Let A₁, A2,..., An be a finite sequence of pairwise disjoint sets in F
with corresponding measures µ(A₁), µ(A₂), ...‚µ(An). Prove that for any measurable
set EC X, the following holds:
µ (E^Ư²²_₁ ¹₂) = Σï²–₁ µ(E Ñ A;)
Transcribed Image Text:Let (X, F, μ) be a measurable space, where X is a set, F is a sigma-algebra on X, and μ is a measure on F. Let A₁, A2,..., An be a finite sequence of pairwise disjoint sets in F with corresponding measures µ(A₁), µ(A₂), ...‚µ(An). Prove that for any measurable set EC X, the following holds: µ (E^Ư²²_₁ ¹₂) = Σï²–₁ µ(E Ñ A;)
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