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Consider measurable spaces $(A_1,\mathcal{A}_1)$, $(A_2,\mathcal{A}_2)$ and their product $\sigma$-algebra $(A_1\times A_2,\mathcal{A}_1\otimes \mathcal{A}_2)$. I want to show, that

$$f:A_1\times A_2\longrightarrow\mathbb{R}$$

is measurable. Is it sufficient, to show that for any $a_1\in A_1$, $a_2\in A_2$ the functions

$$f(\cdot,a_2):A_1\longrightarrow\mathbb{R}$$

$$f(a_1,\cdot):A_2\longrightarrow\mathbb{R}$$

are measurable?

J. W. Tanner
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user408858
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    It is not sufficient, see David Mitra's comment to the question https://math.stackexchange.com/q/647235/27978 (the book referenced, Gelbaum and Olmsted, "Counterexamples in Analysis" is well worth having in your library if real analysis is your thing). – copper.hat Jan 23 '19 at 17:58

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