Sattelhofsk

2022-06-21

MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document We have the outer measure

The inner measure is:

Where ${\mu }_{0}\left(A\right)=\sum _{n=1}^{\mathrm{\infty }}{\mu }_{0}\left({A}_{n}\right)$ and ${\mathcal{A}}_{0}$ is a $\sigma$ algebra.
But why is ${\mu }_{\ast }\le {\mu }^{\ast }$?
I always end up with a contradiction.
My proof would be:

Because:

We get:

Because by definition ${\mu }^{\ast }\left(A\right)=inf\sum _{n=1}^{\mathrm{\infty }}{\mu }_{0}\left({A}_{n}\right)\le \sum _{n=1}^{\mathrm{\infty }}{\mu }_{0}\left({A}_{n}\right)$ this means, that ${\mu }_{0}\left(X\mathrm{\setminus }A\right)-{\mu }^{\ast }\left(X\mathrm{\setminus }A\right)\ge 0$
We get:

So where's my mistake?

gaiageoucm5p

${\mu }^{\ast }\left(A\right)={\mu }_{0}\left(A\right)-{\mu }_{\ast }\left(\mathrm{\varnothing }\right)$ is incorrect. We have ${\mu }_{0}\left(A\right)-{\mu }_{\ast }\left(\mathrm{\varnothing }\right)={\mu }_{0}\left(A\right)-{\mu }_{0}\left(X\right)+{\mu }^{\ast }\left(X\right)$.