X be a set, $\sum$ be any $\sigma$- algebra on X, is there a measure on X such that every element of $\sigma$- algebra is measurable? In particular, on $\mathbb{R}$, $2^{\mathbb{R}}$ be a $\sigma$- algebra, is there a measure on $\mathbb{R}$ such that every element of $2^{\mathbb{R}}$ is measurable?

Pick any $x \in X$ and define $\mu(A)=1$ if $x \in A$, $\mu(A)=0$ for $x \notin A$. This is a measure and every set is measurable. This works for any set and any sigma algebra on it. 
