# 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 $$\epsilon$$. The Rudin’s proof of the Riesz representation theorem construct measure $$\mu$$ and measurable set $$\mathfrak{M}$$, then prove the $$\mu$$ and $$\mathfrak{M}$$ 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. $$\epsilon$$ changes left and right parts of inequality (bidirectional inequality).

If $$\Sigma^n_1\mu(K) \le \Sigma^n_1\mu(V_i)$$ then $$\Sigma^{\infty}_1 \mu(K) \le \Sigma^{\infty}_1 \mu(V)$$, K is compact and V is open. We can change both sides of inequality with $$\epsilon$$. $$\Sigma^{\infty}_1 \mu(V) \le \Sigma^{\infty}_1 \mu(K) + \epsilon$$.

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

##### Jun Kang
###### Clinical Assistant Professor of Hospital Pathology

My research interests include pathology, oncology and statistics.