# Proof of lebesuge measurable function

If f : Rn -> R is Lebesgue measurable on Rn, prove that the function F : Rn * Rn -> R de fined by F(x, y) = f(x - y) is Lebesgue measurable on Rn * Rn.

how can I prove this question?

You should provide us some detail about what attempts you've made. From what you've given, I'm not sure what you're stuck on and how I can help. I'll assume you've tried the obvious thing, which is look at the definition, F is measurable if $$F^{-1}(a,\infty]$$ is measurable. Since f is measurable, we know $$f^{-1}(a,\infty]$$ is measurable. Now try to relate the points in this set to the points in the former set. It might help to look at it visually. Now, using this process will get you thinking about the function in the right way, but there are easier ways to prove it. You might want to think about composing measurable functions.
You should provide us some detail about what attempts you've made. From what you've given, I'm not sure what you're stuck on and how I can help. I'll assume you've tried the obvious thing, which is look at the definition, F is measurable if $$F^{-1}(a,\infty]$$ is measurable. Since f is measurable, we know $$f^{-1}(a,\infty]$$ is measurable. Now try to relate the points in this set to the points in the former set. It might help to look at it visually. Now, using this process will get you thinking about the function in the right way, but there are easier ways to prove it. You might want to think about composing measurable functions.