Closure of a measurable set from $[0,1]$ when it has measure 1.

Question

Suppose $E \subseteq [0,1]$ is measurable sets in measure space $(X,\Sigma,\mu)$ and $\mu(E)=1$. Prove $\bar{E}=[0,1]$.

Answer 1

Suppose that $x\notin\overline{E}$.

Then an interval $[a,b]\subseteq[0,1]$ will exist (that contains $x$) that has an empty intersection with $E$.

This leads to $\mu(E)\leq1-(b-a)<1$