Limit of inequality of sequence and epsilon

Here I summarize some tools for proof of the Riesz representation theorem. They are the limit of inequality of sequence and $ϵ$. The Rudin’s proof of the Riesz representation theorem construct measure $\mu$ and measurable set $M$, then prove the $\mu$ and $M$ have properties. Countable additivity (not subadditivity) is an important property. The strategy of proving equality (additivity) is bidirectional inequality.

Limit of inequality of sequence gives us a tool that finite inequality makes infinite inequality. $ϵ$ changes left and right parts of inequality (bidirectional inequality).

If ${\Sigma }_1^n\mu \left(K\right)\le {\Sigma }_1^n\mu \left(V_i\right)$ then ${\Sigma }_1^{\infty }\mu \left(K\right)\le {\Sigma }_1^{\infty }\mu \left(V\right)$, K is compact and V is open. We can change both sides of inequality with $ϵ$. ${\Sigma }_1^{\infty }\mu \left(V\right)\le {\Sigma }_1^{\infty }\mu \left(K\right)+ϵ$.

Urison’s lemma is used for $\mu \left(E\right)=inf\mu \left(V\right)=sup\mu \left(K\right)$ if $E$ and $KEV$.