Measure theory as it relates to Cumulative distribution functions (CDFs), working on problem sets.

## Moar Measure

Following up on my study of measure theory, which in turn, is a study of probability theory because a probability is a measure, here are some properties of a measure $\mu$ (via the first few videos of the mathematical monk's playlist on probability):

- Monotonicity: $A \subset B \implies \mu(A) \leq \mu(B)$
- Subadditivity: $E_1, E_2, ... \in A \implies \mu(\bigcup_{i}E_i) \leq \sum_{i}\mu(E_i)$
- Continuity from below: $E_1, E_2, ... \in A$ and $E_1 \subset E_2 \subset ... \implies \mu(\sum_{i=1}^{\infty}E_i) = \lim_{i\to\infty} \mu(E_i)$
- Continuity from above: if $E_1, E_2, ... \in A$ and $E_1 \supset E_2 \supset ... $ and $\mu(E_1) < \infty$ then $\mu(\bigcap_{i=1}^{\infty} E_i) = \lim_{i\to\infty} \mu(E_i)$

The above are true of all measures, and thus all probability measures. Here are some more properties of probability measures.

Let $(\Omega, \mathcal{A}, P)$ be a probability measure space with $E, F, E_i \in \mathcal{A}$

- $P(E \cup F) = P(E) + P(F)$ if $E \cap F = \emptyset$
- $P(E \cup F) = P(E) + P(F) - P(E \cap F)$
- $P(E) = 1 - P(E^C)$
- $P(E \cap F^C) = P(E) - P(E \cap F)$

The generalization of $P(E \cup F) = P(E) + P(F) - P(E \cap F)$ is called the inclusion exclusion principal and can be visualized with 3 sets:

All of these are enumerated as properties of probability measures in the all of stats book, but I don't mind running through it again with my new found appreciation for measure theory. Most of this stuff can be visualized with venn diagrams.

## Borel Probability measures and CDFs

Let's consider a Borel measure on $\Bbb R$, which is a measure with $\Omega$ is $\Bbb R$, $\mathcal{A}$ is $\mathcal{B}(\Bbb R)$, e.g it is a measure on $(\Bbb R, \mathcal{B}(\Bbb R))$.

$\mathcal{B}(\Bbb R))$ is the Borel $\sigma$-algebra which are all of the open sets of $\Bbb R$.

Going back to cumulative distribution functions (CDFs), it turns out that every CDF implies a unique Borel probability measure and that a Borel probability measure implies a unique CDF.

More concretely, A CDF is defined as a function $F: \Bbb R \to \Bbb R$ s.t.

- $x \leq y \implies F(x) \leq F(y)$ $(x,y \in \Bbb R)$ ($F$ is non-decreasing)
- $lim_{x \searrow a} F(x) = F(a)$ ($F$ is right-continuous)
- $lim_{x \to \infty} F(x) = 1$
- $lim_{x \to -\infty} F(x) = 0$

Theorem: $F(x) = P((-\infty, x])$ defines an equivalence between CDFs $F$ and Borel probability measures $P$.

This is kind of interesting as it frames the scope of what kind of probability measures can be uniquely described with a CDF: those that are measures over the Borel $\sigma$-algebra, e.g those considering the open sets of real numbers $\Bbb R$.

So this means we might in some cases be trying to reason about probability measures that are *not* on open sets of real numbers, and we'll need to use measure theory as a CDF won't cut it.

## Mathematical monk's recommended resources

In this video the teacher recommends some books, so I thought I'd take note in case I need even moar stuff to read / reference / study:

- Rudin's principles of mathematical analysis
- Jacod Protter probability essentials
- Real Analysis: Modern Techniques (advanced)

all are googleable to find excerpts / problem sets.

## Useful tool for looking up LaTeX symbols

As I attempt to get more fluent in typing out TeX for Mathjax, being able to find a symbol quickly is important. This tool rocks: it lets you draw the symbol and finds closely related symbols along with the needed TeX.

## Starting on problem sets

I'm finally diving into the All of Stats problem sets from the course website, here's my WIP for problem 1.