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For spaces $X$ and $Y$ with measures there is an isomorphism of Hilbert spaces $$L^2(X) \otimes L^2(Y) \to L^2(X\times Y), ~~ f\otimes g\mapsto \left((x,y)\mapsto f(x)g(y)\right).$$

Now suppose $E \to X$ is a (locally trivial) fiber bundle with fiber $F$. Is there any nice relationship between $L^2(E)$ adn $L^2(F)\otimes L^2(X)$?

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