Let $\gamma : [0,1] \rightarrow \mathbb{R}^{2}$ continuous function such as $\gamma(0)=\gamma(1)$. Let $\rho_{n}$ unit mollifier. Let, for $n\geq 1$, $A_{n}:=\gamma \star \rho_{n}([0,1])$ and $A=\gamma ([0,1])$.
Is it true that $$\lim_{n }\int \chi_{A_{n}}(x)dx=\int \chi_{A}(x)dx.$$ $\chi$ is the indicator function.
I would like to use the TCD but I can't prove the pointwise convergence.