# Doubt in proof of Riesz representation theorem for $L^1$

I understand everything except 2 doubt

1) Why u is measurable

2) why $$u\chi_n\in L^2(\Sigma)$$

I have highlighted text.

Any Help will be appreciated

Quotient of two measurable functions is still measurable, this follows by the composition of the continuity map $$(u,v)\rightarrow u/v$$ and $$x\rightarrow(f(x),g(x))$$ for measurable functions $$f,g$$.
On the set $$\Omega_{n}$$, $$\theta^{-1}\leq\epsilon_{n}^{-1}$$, so $$\|u\chi_{\Omega_{n}}\|_{L^{2}}\leq\epsilon_{n}^{-1/2}\|v\|_{L^{2}}$$.