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?
• 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 '18 at 18:46