Decompose Measurable Set into Two Measurable subsets

Question

Let $E$ be a Lebesgue measurable set (with positive measure).

Under what suitable conditions can we write $E=A\cup B$, where $A$ and $B$ are measurable with positive measure?

What I understand is that this is not always true for arbitrary decompositions, for instance $E=[0,1]$ can be decomposed into the Vitali Set and its complement (which are both non-measurable).

Thanks for any help.

Some other trivial examples I can think of are:

• $E$ is an interval, can be decomposed into two intervals
• $E$ is already known to be the union of two measurable sets with positive measure

Answer 1

Suppose $E$ is a Lebesgue-measurable subset of $\mathbb R$ with $\mu(E)\gt0.$ Then the function $f(x)=\mu(E\cap(-\infty,x])$ is continuous, so it assumes all values between $f(-\infty)=0$ and $f(+\infty)=\mu(E).$ Choose $x$ with $f(x)=\frac12\mu(E)$ and decompose $E$ into the sets $A=E\cap(-\infty,x]$ and $B=E\cap(x,+\infty).$

P.S. The function $f(x)$ is continuous because $|f(x)-f(y)|\le|x-y|.$