real analysis - A question on convergence to zero of measurable sets

Let $f$ be a strictly positive (almost everywhere) measurable function which is also integrable. Let $E_n$ be a sequence of measurable sets such that $\int_{E_n}f\to 0$ as $n\to\infty.$ Is it true that $\mu(E_n)\to 0$ where $\mu$ is a positive measure with respect to $f$ is integrable. The measure space is finite.