Let E have measure zero Show that if f is a bounded function

Let E have measure zero. Show that if f is a bounded function on E, then f is measurable and Integral_E f = 0.

Solution

the solution is very simple

Let F be a measurable subset of R. Then f^-1(F) is a subset of E(since F is bounded on E), and since subsets of measure zero sets have measure zero, f¹(F) has measure zero. In particular f¹(F) is measurable, so f is measurable