The discussion revolves around proving that the function F : Rn * Rn -> R, defined by F(x, y) = f(x - y), is Lebesgue measurable given that f : Rn -> R is Lebesgue measurable. The scope includes mathematical reasoning and proofs related to measure theory.
Participants do not appear to have reached a consensus, as there are multiple approaches suggested and requests for clarification on the proof process.
Participants express uncertainty about the relationship between the sets involved in the proof and the methods to establish measurability, indicating that further exploration of definitions and properties of measurable functions is needed.
ebola1717 said: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.