Measures and Probability Measures

Let \((\Omega, \mathcal{F})\) be a measurable space (see Sigma Field). A measure is a function \(\mu : \mathcal{F} \rightarrow [0, \infty]\) such that:

1. μ(∅) = 0
2. (countable additivity or \(\sigma\) additivity) if \(\mathcal{A_i}\) is a (countably infinite) sequence of disjoint sets, then \(\mu(\cup_{i} A_i) = \sum_i \mu(A_i)\)

A probability measure \(\mathbb{P}\) is a measure with the additional property that \(\mathbb{P}(\Omega) = 1\). In that case \((\Omega, \mathcal{F})\) together with \(\mathbb{P}\) is called a probability space.

If an event \(A\) has \(\mathbb{P}(A) = 1\), we say that \(A\) occurs almost surely. (For comparison see Convergence Almost Surely). Note that this does not mean that \(A = \Omega\). There could be other events in \(\mathcal{F}\) that have 0 probability.

Probability measures have the following properties

1. (finite additivity) If the events \(A_1,...,A_n\) are disjoint, then \(\mathbb{P}(\cup_i^n A_i) = \sum_{i=1}^{n} \mathbb{P}(A_i)\)
2. For any event \(A\), we have \(\mathbb{P}(A) = 1 - \mathbb{P}(A^C)\)
3. For events \(A\) and \(B\), if \(A \subset B\), then \(\mathbb{P}(A) < \mathbb{P}(B)\)
4. (union bound) For any sequence \(\{A_i\}\) of events, we have \[ \mathbb{P}\left(\cup_{i=1}^{\infty} A_i \right) \leq \sum_{i=1}^{\infty} \mathbb{P}(A_i) \]

5. (inclusion-exclusion formula) For any collection of events \(A_1, ..., A_n\)

   \begin{align*} \mathbb{P}\left(\cup_{i=1}^{\infty} A_i \right) = & \sum_{i=1}^n\mathbb{P}(A_i) - \sum_{(i,j): i < j} \mathbb{P}(A_i \cap A_j)\\ & + \sum_{(i,j,k): i < j < k} \mathbb{P}(A_i \cap A_j \cap A_k) \\ & + ... + (-1)^{n-1} \mathbb{P}(A_1 \cap ... \cap A_n) \end{align*}

Proof for the inclusion exclusion formula using indicator random variables and expectation can be found in these lecture notes.

• Lebesgue Measure