Suppose $f$ be measurable and $\int_{I}fd\lambda=0$ for all bounded intervals $I$ in $\mathbb{R}$ and $\lambda$ is the Lebesgue measure. Then, is $\lambda(\{x\in\mathbb{R}:f(x)\neq0\})=0$?
I think yes, because, $\int_{I}fd\lambda=\int_0^k\lambda(\{x:|f(x)|>t\})dt$ where $I$ be the intervals $[0,k]$. Since the terms are all zero, so we can conclude that $\lambda(\{x\in\mathbb{R}:f(x)\neq0\})=0$. Any hints? Thanks beforehand.