I am (self-)studying the book by Rosenthal called A first look at rigorous probability theory. My question is on verifying the conditions on a probability measure $\mathbb P$ of the Uniform[0,1] distribution such as to apply the Extension Theorem (Theorem 2.3.1, page 10). More specifically, I would like to solve Exercise 2.4.3 on page 15. Definitions and variables used are all explained on page 15.
Let me highlight that ``interval'' is understood to include all the open, closed, half-open, and singleton intervals contained in $[0,1]$, and also the empty set $\emptyset$.
(a) Let $a_j$ be the left end-point and $b_j$ the right end-point of the interval $I_j$ and similary $a_0$ and $b_0$ for the interval $I$. By re-ordering we can ensure $a_0 \ge a_1 \le b_1 = a_2 \le \ldots \le b_k \ge b_0$. Thus, \begin{align} \sum_{j = 1}^n \mathbb P(I_j) = \sum_{j = 1}^n (b_j - a_j) = b_k - a_1 \ge b_0 - a_0 = \mathbb P (I). \end{align}
(b) $I_1,I_2,\ldots$ is a countable collection of open intervals with $\bigcup_{j=1}^\infty I_j \supseteq I$ for some interval $I$. Using the Heine-Borel Theorem we have that $\exists k : \bigcup_{j = 1}^k I_j \supseteq I$. How to continue from here?
(c) I do not know how to solve this part.
Rosenthal
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