Can anyone help with this proof, please(Measure Theory)?

Question

Let A be a subset of R^n with Lebesgue measure m(A) < oo. Let x + A be the translation of A by the vector x of R^n and define f:R--> [0, oo) by f(x) = m( A intersect (x + A)). Show that f is continuous.

My idea was to prove this first to nice sets like cells and then approximate A by a sequence of such nice sets. I tried to obtain a sequence of continuous functions converging uniformly to f, which would establish its continuity. But I got mixed up.

Is this aproach good? Can anyone help, maybe suggesting a different approach.

Thank you

Correcting a typo: On the definition of f, it is f^R^n --> [0,oo), of course

Answer 1

I have a feeling that |m(A∩(X+A) - m(A∩(x+A))| < ε whenever |x-X| < ε^n/mA ... I think this is pretty easy to show for rectangles, and should hold true for measurable sets in general as well... Haven't had time to try to prove it yet, though.

Assuming this is true, the problem is done, of course, since you then have your δ (=ε^n/mA) for any given ε.

Answer 2

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