Assume that $(X,\Sigma)$ and $(Y,\text{T})$ are measurable spaces and the function $f:X\times Y\to\mathbb{R}$ is not measurable. Can we say anything about the (non) measurability of the function $f(x,\cdot):Y\to\mathbb{R}$ for fixed $x\in X$?
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1Take $f(x,y)=g(y)$ where $g$ is not measurable. – Kavi Rama Murthy Jun 14 '21 at 12:06
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@KaviRamaMurthy oh thanks! If we exclude this case, can we then say anything about the (non) measurability? – Nicki Jun 14 '21 at 12:20
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1No. $f(x,.)$ can be measuarble for each $x$ without $f$ being measurable. – Kavi Rama Murthy Jun 14 '21 at 12:25