# Measurable set proof

1. Aug 26, 2014

### mahler1

1. The problem statement, all variables and given/known data

Let $E \subset \mathbb R^n$ be a measurable set such that $E=A \cup B$ with $|B|=0$ ($B$ is a null set). Show that $A$ is measurable.

3. The attempt at a solution

I know that given $\epsilon$, there exists a $\sigma$-elementary set $H$ such that $E \subset H$ and $m_e(H-E)<\epsilon$. How can I construct a $\sigma-$elementary set $H'$ such that $m_e(H-A)<\epsilon$?. Any suggestions would be appreciated

Last edited: Aug 27, 2014
2. Aug 27, 2014

### HallsofIvy

Staff Emeritus
What does |B| mean? Is it the measure of B or is the cardinality of B (in which case B is empty so that E= A).

In your last sentence do you not mean "How can I construct an elementary set H' such that $m_e(H'- A)<\epsilon$"?

3. Aug 27, 2014

### mahler1

Edited, thanks for the correction.