Proving Measurability of ##A## from ##E=A \cup B## with ##|B|=0##

[mahler1]

Homework Statement

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.

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

 

[HallsofIvy]

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 $m_e(H'- A)<\epsilon$"?

 

[mahler1]

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 $m_e(H'- A)<\epsilon$"?

Edited, thanks for the correction.