Suppose you have a function $f\in L^p(\mathbb{R}^n)$ and some bounded set $\Omega$ of measure 1. Define $$ f_\epsilon = \frac1{\epsilon^n}\int_{\Omega_\epsilon} f(x + y)dy $$ Where $\Omega_\epsilon$ is the set $\{y:y/\epsilon\in\Omega\}$. My problem is to prove that $f_\epsilon\to f$ in $L^p$.
It has been several years since I've taken an analysis class and I recall solving problems like this without too much trouble, but it seems like I've forgotten something crucial. I've tried various applications of Jensen's Inequality, Holder's inequality, Fubini's Theorem, rewriting $f_\epsilon$ as a convolution, and clicking through related topics on Wikipedia, but nothing I do seems to be leading toward a solution.
This isn't homework, it came from an old analysis final at my school.
I hope this is not a duplicate, it is a very TeX-y question and difficult to search for. Apologies if it is completely trivial.
Let $\tau_h$ denote the translation operator, given by $\tau_h f(x) = f(x+h)$. Now, for $1 \leqslant p < \infty$, we know that $C_c(\mathbb{R}^n)$ is dense in $L^p(\mathbb{R}^n)$, and it is easy to see that for $f \in C_c(\mathbb{R}^n)$ we have $\lim\limits_{h\to 0} \lVert \tau_hf - f\rVert_p = 0$. Since the $\tau_h$ are isometries, the family $\{\tau_h : h \in \mathbb{R}^n\}$ is an equicontinuous family of operators $L^p \to L^p$, hence we have $\lim\limits_{h\to 0} \lVert \tau_h f - f\rVert_p = 0$ for all $f \in L^p$ (even uniformly on all compact subsets of $L^p$, but we don't need that).
Now, we can also write (by a change of variables $y' = \varepsilon y$ in the definition of $f_\varepsilon$)
$$f_\varepsilon(x) = \int_\Omega f(x+\varepsilon y) \, dy,$$
and then we can estimate $\lVert f_\varepsilon - f\rVert_p$ by an average of $\lVert \tau_hf -f\rVert_p$:
$$\begin{align} \lVert f_\varepsilon - f\rVert_p^p &= \int_{\mathbb{R}^n} \left \lvert \int_\Omega f(x+\varepsilon y) - f(x)\, dy \right\rvert^p\,dx \quad \text{(Jensen)}\\ &\leqslant \int_{\mathbb{R}^n} \int_\Omega \lvert f(x+\varepsilon y) - f(x)\rvert^p \,dy\,dx \quad \text{(Fubini)}\\ &= \int_\Omega \int_{\mathbb{R}^n} \lvert f(x+\varepsilon y) - f(x)\rvert^p\,dx\,dy\\ &= \int_\Omega \lVert \tau_{\varepsilon y}f - f\rVert_p^p\,dy. \end{align}$$
Since $\Omega$ is bounded, by the above continuity property of the translations, for every $\delta > 0$ we can find an $\eta > 0$ such that for all $0 < \varepsilon < \eta$ and all $y \in \Omega$ we have $\lVert \tau_{\varepsilon y}f - f\rVert_p < \delta$, and that implies
$$\lim_{\varepsilon \to 0} \lVert f_\varepsilon -f \rVert_p = 0.$$
For $p = \infty$, the assertion does not hold.