Proof clarification of the separability of L^p

Question

I'm reading Billingsley's Probability and Measure. I'm having trouble understanding the proof of the following theorem 19.2 part (ii).

I don't understand what is highlighted in yellow.

i) Why is $\mathcal{F}_n$ an algebra (field). In particular, why is it closed under finite unions and complements?

ii) Why does $\mathcal{F}_0$ generates $\mathcal{F}$?

iii) Why is $\mu$ sigma finite on $\mathcal{F_0}$?

iv) I dont understand when it says "any $\alpha_i$ that vanishes can be suppressed".

I'll really appreciate some help. I've already tried things but without results.

Answer 1

1. The collection of sets of the form $F_1\cap\dots\cap F_n$ with $F_i=E_i$ or $E_i^c$ can be expressed as $A_1,\dots,A_{2^n}$ , where the sets $A_i$ are pairwise disjoint and their union is $\Omega$. A set in $ \mathcal F_n$ can be expressed as $\bigcup_{i\in I}A_i$ for some subset $I$ of $\{1,\dots,2^n\}$ and using the fact that the $A_i$ are disjoint $$ \bigcup_{i\in I}A_i\cup\bigcup_{j\in J}A_j=\bigcup_{i\in I\cup J}A_i,\quad \left(\bigcup_{i\in I}A_i\right)^c=\bigcup_{i\in I^c}A_i, $$ which clearly belongs to $\mathcal F_n$.
2. This is due to the fact that $\mathcal F_0$ contains for each $n$ the sets $E_1,\dots, E_n$.
3. Because $\mathcal F_0$ contains $\mathcal D$.
4. At some point, one needs to divide by $\lvert \alpha_i\rvert$ in order to get $\varepsilon$ and not $\lvert \alpha_i\rvert$. This is possible only if $\alpha_i\neq 0$ but we can reorder the $A_i$'s and ignore the terms for which $\alpha_i=0$, giving a representation of the form $g=\sum_{i=1}^{m'}\alpha'_i\mathbf{1}_{B'_i}$ with $\alpha'_i\neq 0$ for each $i$.