# Understanding Borel Sets in Relation to Distributions

Following up with my last question, I am self studying the book Elementary Stochastic Calculus with Finance in View. The author has lost me in some of the terseness of his explanation of distributions. I think some of it is the author sparing the reader the gorey theoretical details.

The author goes on to define the collection of probabilities

$P_x(B) = P(X \in B) = P(\{\omega : X(\omega) \in B\})$ for "suitable" subsets $B \subset \!R$

as the distribution of X. Intuitively I feel like this defines a Probability Mass Function for the random variable X. I am having trouble reasoning this out in my head.

He continues, stating the "suitable" subsets $B$ of $\!R$ are the Borel Sets, and that they are obtained by a countable number of operations $\cap, \cup, ^c$ acting on the intervals.

He does not specify which intervals. He uses this to hand-wave justify the equivalence of $F_x$, the so called Distribution Function (AKA the CDF) and this new concept, the Distribution. Through this equivalence either of them can be used calculate $P(X \in B)$.

How are borel sets related to probability and how do they help prove the equivilence of the CDF and PMF of a distribution? Part of my problem is thinking about this new concept of Borel Sets related to a random variable $X$. What does it contain? Is there a concrete example that can help solidify this in my head? I was able to understand $\sigma$-algebras, outcome spaces, and random variables being used to create a probability measure by looking at a simple coin flipping example. Does something like this exist to "teach by example" what a borel set is and it's relation to a probability distribution?

Thank you!

• Re "which intervals:" by definition, an "interval" is the intersection of two half lines of the form $[b,\infty)=\{x\in\mathbb{R}\mid x \ge b\}$ and $(-\infty, a)=\{x\in\mathbb{R}\mid x \lt a\}.$ – whuber Aug 6 '18 at 12:37