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 μ and measurable set M, then prove the μ 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 Σn1μ(K)≤Σn1μ(Vi) then Σ∞1μ(K)≤Σ∞1μ(V), K is compact and V is open. We can change both sides of inequality with ϵ. Σ∞1μ(V)≤Σ∞1μ(K)+ϵ.
Urison’s lemma is used for μ(E)=infμ(V)=supμ(K) if E and KEV.
