# 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.

TODO