Proving the Measurability of a Function Composition

[justin_huang]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

I am confused by the expression of F(x,y), it seems x-y is equal to x of f(x), how can I prove this question?

 

[micromass]

You know that $$\mathbb{R}^n\times\mathbb{R}^n\rightarrow \mathbb{R}^n:(x,y)\rightarrow x-y$$ is continuous and thus measurable right?

The only thing you need to show then is that the composition of measurable functions is measurable...