1. Oct 3, 2013

Cascabel

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?

2. Oct 3, 2013

Cascabel

Sorry, figured it out.

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