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?

Question

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?

Answer 1

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.

Answer 2

This is not a complete answer, but only treats the case mentioned in the body of your question.

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Yes.

E.g. let the measure be prescribed by $A\mapsto|A|$ if $A$ is finite and $A\mapsto+\infty$ otherwise.

There are more (see the comment of Gabriel on your question).