Another question about integrable functions with a transform I am an engineering student, and taking a real analysis course at demand of my advisor, my inexperience in proofs is giving me hard time. I stumbled upon this example, whose proof left as an exercise. Could you please give me some clues? Here is the example:
$f\in L^1(\mathbb R^n).$ Show that
$$
\lim_{t\rightarrow 1} \int_{\mathbb R^n}|f(x)-t^nf(tx)|\; dm=0
$$ 

Here is a lacunary solution.


*

*Approximate $f$ in $\mathbb L^1$ by a simple function (a linear combination of characteristic functions of measurable sets: this reduced the proof where $f$ is such a function. 

*By linearity, do it when $f=\chi_B$, where $B$ is a Borel subset of $\mathbb R^n$, with finite Lebesgue measure. 

*By outer regularity, we are reduced to threat the case $B$ open. 

*Use a dominated convergence argument. 

