# Measure theory question: Countable sub-additivity

I have a question on sub-additivity. For sets ##E## and ##E_j##, the property states that if

##E=\bigcup_{j=0}^{\infty}E_j##

then

##m^*(E) \leq \sum_{j=0}^{\infty}m^*(E_j)##, where ##m^*(x)## is the external measure of ##x##.

Since ##E\subset \bigcup_{j=0}^{\infty}E_j##, by set equality, the property seems to follow from monotonicity.

However, it is also true that, ##\bigcup_{j=0}^{\infty} E_j \subset E##, which seems to imply the reverse inequality, ##\sum_{j=0}^{\infty} m^*(E_j)\leq m^*(E)##, which is not true.

What's wrong?

## Answers and Replies

Sorry, figured it out.

Sorry, figured it out, ##m^* \left (\bigcup_{j=0}^{\infty}E_j \right ) \neq \sum_{j=0}^{\infty} m^*(E_j)##.