Proof of lebesuge measurable function

[justin_huang]

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?

 

[ebola1717]

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.

 

[justin_huang]

how to relate the points these two set? could you please give me more detailed method?

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.

 

[Eynstone]

Observe that the family of lines {x-y= constant} chops the domain into measurable 'slices'.
You may also try as suggested by ebola1717 and prove a general result
about F(x,y) = f(g(x,y)) where g is measurable.