How to show that the limit of the following integrals is zero?

Question

Let $f:[0,1]\longrightarrow [0,\infty]$ be a Lebesgue measurable function such that $\int_{[0,1]}f dm<\infty$. I would like to show that $\displaystyle\lim_{c\longrightarrow 0^+}\int_{[0,c]}f dm=0$ (where $c\in (0,1]$). Monotone Convergence Theorem or Dominated Convergence Theorem might be helpful, but we cannot apply these theorems directly. Does anybody have an idea ?

Answer 1

Assuming the given conditions in the post and applying the comments, one defines $g_n=f\chi_{[0,\frac 1n]},$ where $\chi$ is the indicator function. Clearly $g_n\leq f$ and $\lim_{n\rightarrow\infty}g_n=0,~{\rm a.e.,}$ hence by dominated convergence theorem, one has $$\lim_{n \rightarrow\infty}\int_{[0,1]}g_n~dm =0.~\qquad (1)$$ Now for any $c$ with $0<c<\frac 1 n$, one has $$\int_{[0,c]}f~dm=\int_{[0,1]}f\chi_{[0,c]}~dm\leq \int_{[0,1]}g_n~dm.$$ It follows from (1) that $$\lim_{c\rightarrow 0^+}\int_{[0,c]}f~dm=0.$$