# Probability Measures

The third component of a probability space is a probability
measure \(P\) on the events in the event space \(\mathscr{F}\). A
*measure* is just a mathematical function that assigns measurements to things being measured. For us,
this means assigning probabilities to events. So \(P(A)\) is the probability of the event \(A\), where
\(A \in \mathscr{F}\). That's really all we need to know here about how measures work.

But besides \(P\) being a measure, we also need \(P\) to behave in a way that comports with our expectations
around how probabilities work. There are three foundational principles, or *axioms*, that \(P\) must
satisfy to be a good *probability* measure. So let's state those.

## Axioms of probability

**Axiom 1:** \(P(A) \geq 0\) for all \(A \in \mathscr{F}\). That is, a probability is defined
for every event \(A \in \mathscr{F}\), and it's nonnegative.

**Axiom 2:** \(P(\Omega) = 1\). Intuitively, the probability for the whole
sample space is 1, which means that when we run an experiment, at
least one of the outcomes must occur.

**Axiom 3:** \(P(\bigcup_{i \in J}{A_i}) = \sum_{i \in J}{P(A_i)}\) for countable index set
\(J\) and disjoint \(A_i\). This one is a little trickier to understand, but not too bad. For countably many
\(A_i\), the probability of their union is just the sum of their individual probabilities. Which makes
sense.

Now these are axioms, so that means that they're intended as an minimalistic/economical way to state the
requirements on \(P\). But there some other factoids—factoids that follow more or less immediately from
the axioms—that you'll need to know. We'll call these *key properties* for lack of a better
term.

## Key properties of probability

TODO