Suppose $f(x,y)$ is a function on $\mathbb{R}^2$ such that:

1) For a fixed $x$, $f(x,y)$ is a measurable function of $y$ and 2) For a fixed $y$, $f(x,y)$ is a continuous function of $x$.

Prove that for a function $g(y)$ that is measurable on $\mathbb{R}$, we have that $f(g(y),y)$ is measurable on $\mathbb{R}$.

I can prove that f(x,y)is measurable on $\mathbb{R}^2$,but I don't know how to prove this problem?

For f(x,y) is just measurable on $\mathbb{R}^2$,is the result still right?

  • Welcome to MSE. Please read this text about how to ask a good question. – José Carlos Santos Dec 7 at 9:37
  • You say that you know how to prove that $f(x,y)$ is measurable. Also $y\mapsto (g(y),y)$ is measurable, right? What happens when you take the composition? – Federico Dec 7 at 18:46

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