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Let $X$ be a subset of $[0,1]$ with length (Lebesgue measure) $1$.

For each $x\in X$, there is some $\epsilon_x > 0$. Define $I_x$ as the open interval $(x, x+\epsilon_x)$. I am interested in the union of all these intervals:

$$U := \bigcup_{x\in X} I_x$$

Initially I thought that $U$ contains the entire unit interval, but this is not true. For example, it is possible that for each $x<0.5$, $\epsilon_X := (0.5 - x)/2$. Then, $0.5

ot \in U$.

My quesion is: is the length of $U$ always at least $1$?