Thread: Lebesgue measurable function and Borel measurable function

Let f(x) be a real-valued Lebesgue measurable function on , show that there exist Borel measurable function g and h such that g(x)=h(x) a.e.[m], and .
"a.e.[m]" means " holds almost everywhere with respect to lebesgue measure".

I know this problem is good problem ofr helping us understand the relation between lebesgue and borel measurability. It concerns the completion of Borel measurable set class. And something else.....
Any help will be appreciate.

this problem is difficult. I am open to any idea or solution.
My idea:
lebesgue measurable set can be written as the Union of a borel measurable set and a set with measure 0.
and the Lemma 9 and proposition 10 in Page 260-261 may help us. (see Royden's Real Analysis.)