Let $f: [0, 1] \to \mathbb R$ be Lebesgue integrable with $\int_0^1 f \, d \mu = C.$

**Question:** For every $K$ with $0 < K \leq 1$, does there exist an open subset $U$ of $[0, 1]$ of Lebesgue measure $K$ such that

$$\frac{1}{K} \int_U f \, d\mu = C?$$

*Remark:* This is relatively easily seen to be true if $U$ is not required to be open. However I am not sure about the open case.