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$.
