Prove that there exists an uniformly continuous function $g$ such that $f = g$ a.e

Suppose $f \in L^{\infty}(\mathbb{R})$, $f_{h}(x) = f(x+h)$ such that 

$\lim_{h \rightarrow 0} ||f_{h} - f||_{\infty} = 0$ 

Prove that there exist an uniformly continuous function $g$ on $\mathbb{R}$ such that $f =g$ a.e

이것에 대하여 올라온 답변은 다음과 같습니다. 

http://math.stackexchange.com/questions/587748/prove-that-there-exists-an-uniformly-continuous-g-such-that-f-g-a-e/588278