Let $f$ be Lebesgue integrable and continuous in a bounded domain $\Omega$ in $\mathbb{R} ^n$, and define $f_m(x)=f(\frac{m}{m+1}x).$ Then $f_m(x)$ converges pointwise to $f(x)$ for $x\in \Omega.$ I was wondering if it is true $\int_\Omega f_m(x)dx \to \int_\Omega f(x)dx$ as $m\to\infty$, without additional assumption on $f$.
Note that if $f$ is bounded, we can easily show it in view of Lebesgue dominated convergence theorem.
Please let me know if you have any hint or comment for the question. Thanks in advance!
It's very easy to show that $\int_\Omega f_m\to\int_\Omega f$ for every $f\in L^1(\Bbb R^n)$, in fact $||f_m-f||_1\to0$. Let $\epsilon>0$. Choose $g\in C_c(\Bbb R^n)$ with $||f-g||_1<\epsilon$...