Let $f$ be a function on $\mathbb R$ such that $f$ is locally integrable. It is well known that from the Lebesgue differentiation theorem we have $$ \frac{1}{h}\int_t^{t+h} u(s)\,ds \to u(t) $$ almost everywhere if $h \to 0$. My question is, can we prove that the convergence is in $L^1_{loc}$ ?
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Yes or no, depending on what you mean:
Can we prove the convergence is in $L^1_{loc}$, using the almost-everywhere convergence? No, or at least not by any method I know. In particular it does not follow from the almost-everywhere convergence plus DCT; the convergence need not be "dominated".
But yes, we can certainly prove this. In fact it's really just an exercise, as opposed to the almost-everywhere convergence, which definitely counts as a non-trivial theorem.
Exercise. Suppose $f\in L^1(\Bbb R)$, and for $h>0$ define $f_h(x)=\frac 1h\int_0^h f(x+t)\,dt$. Then $||f-f_h||_1\to0$ as $h\to0$.
Hints: (i) Show it's true for $f\in C_c(\Bbb R)$. (ii) Show that the general case follows.
David C. Ullrich
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