Silly Question (monotonic) (updated)

Question

suppose we have a measure $\mu$ on an algebra $B$, and $E,F\in B$

I know, if $E\subseteq F$, then $\mu(E)\leq\mu(F)$. Does the converse true, when $0<\mu(E)\leq\mu(F)$ ($\mu$ is nonatomic).

Answer 1

The converse need not be true. Consider $E$=$[0,\frac{1}{4}]$ and $F$= $(\frac{1}{4},1]$.

$0 < \mu(E)=\frac{1}{4} \leq \mu(F)=\frac{3}{4}$ but $E$ is not a subset of $F$.