Measure, Integration & Real Analysis · Problem 8C.11

Real and Complex Measures — Sheldon Axler

Mathematics — Measure Theory · product-measure-orthonormal-basis

Problem

Suppose

\mu

is a

\sigma

-finite measure on

(X, \mathcal{S})

and

\nu

is a

\sigma

-finite measure on

(Y, \mathcal{T})

. Suppose also that

\{e_j\}_{j\in\Omega}

is an orthonormal basis of

L^2(\mu)

and

\{f_k\}_{k\in\Gamma}

is an orthonormal basis of

L^2(\nu)

for some countable set

\Gamma

. For

j \in \Omega

and

k \in \Gamma

, define

g_{j,k}: X \times Y \to \mathbf{F}

g_{j,k}(x,y) = e_j(x) f_k(y).

Prove that

\{g_{j,k}\}_{j\in\Omega,\, k\in\Gamma}

is an orthonormal basis of

L^2(\mu \times \nu)