Product Sigma-Algebra Generated by Projections

Theorem

Let $\struct {X, \Sigma_1}$ and $\struct {Y, \Sigma_2}$ be measurable spaces.

Let $\Sigma_1 \otimes \Sigma_2$ be the product $\sigma$-algebra on $X \times Y$.

Let $\pr_1: X \times Y \to X$ and $\pr_2: X \times Y \to Y$ be the first and second projections, respectively.

Then:

$\Sigma_1 \otimes \Sigma_2 = \map \sigma {\pr_1, \pr_2}$

where $\sigma$ denotes generated $\sigma$-algebra.

Proof

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Let $R = \{A \times B: A \in \Sigma_1, B \in \Sigma_2\}$ be a generator for the product $\sigma$-algebra $\Sigma_1 \otimes \Sigma_2$.

Note that by Preimage of Element under Projection:

$\pr_1^{-1}(A) = A \times Y \in R$

And:

$\pr_2^{-1}(B) = X \times B \in R$

for any $A \in \Sigma_1$ and $B \in \Sigma_2$.

Let:

$G = \{\pr_i^{-1}(C_i): C_i \in \Sigma_i, i = 1, 2\}$

Thus:

$G \subseteq R$.

However, for any $A \times B \in R$:

$A \times B = \pr_1^{-1}(A) \cap \pr_2^{-1}(B) \in \sigma(G)$

So we have:

$R \subseteq \sigma(G)$.

But:

$G \subseteq R$

and:

$R \subseteq \sigma(G)$

we use Generated Sigma-Algebra Preserves Subset and ($\sigma(G) = \sigma(\sigma(G))$).

Thus:

$\sigma(R) = \sigma(G)$.

It follows from the definition:

$\Sigma_1 \otimes \Sigma_2 = \sigma(R) = \sigma(G) = \sigma (\pr_1, \pr_2)$

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $13.10 \ \text{(i)}$