Measure, Integration & Real Analysis · Problem 5C.3

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · product-sigma-algebras-associativity

Problem

Suppose

(X, \mathcal{S})

(Y, \mathcal{T})

, and

(Z, \mathcal{U})

are measurable spaces. We can define

\mathcal{S} \otimes \mathcal{T} \otimes \mathcal{U}

to be the smallest

\sigma

-algebra on

X \times Y \times Z

that contains

\{A \times B \times C : A \in \mathcal{S}, B \in \mathcal{T}, C \in \mathcal{U}\}.

Prove that if we make the obvious identifications of the products

(X \times Y) \times Z

and

X \times (Y \times Z)

with

X \times Y \times Z

, then

\mathcal{S} \otimes \mathcal{T} \otimes \mathcal{U} = (\mathcal{S} \otimes \mathcal{T}) \otimes \mathcal{U} = \mathcal{S} \otimes (\mathcal{T} \otimes \mathcal{U}).