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Measure & Probability

Basics of Probability

We begin with the sample space $\Omega$ of a random experiment, and a $\sigma$-algebra $\mathcal H$ on $\Omega$ consisting of subsets of $\Omega$ to which we want to assign probabilities.

Definition: A probability measure on $(\Omega, \mathcal H)$ is a function $\mathbb P: \mathcal H \rightarrow [0,1]$ such that

  • $\mathbb P(\emptyset) = 0$
  • If $E_1, E_2, \ldots \in \mathcal H$ is a countable collection of disjoint sets, then $\mathbb P\left( \bigcup E_n \right) = \sum \mathbb P(E_n)$.
  • If $E \in \mathcal H$ then $\mathbb P(E^c) = 1 – \mathbb P(E)$.

The triple $(\Omega, \mathcal H, \mathbb P)$ is called a probability space.

More generally, if $E$ is a set and $\mathcal E$ is a $\sigma$-algebra on $E$, then we call $(E, \mathcal E)$ a measurable space, and a function $\mu : \mathcal E \rightarrow [0, \infty)$ a measure if

  • $\mu(\emptyset) = 0$
  • If $E_1, E_2, \ldots \in \mathcal E$ is a countable collection of disjoint sets, then $\mu\left(\bigcup E_n\right) = \sum \mu(E_n)$.

Measures give a way of ascribing size to sets in $\mathcal E$. The triple $(E, \mathcal E, \mu)$ is called a measure space.

Given $(\mathbb R, \mathcal B(\mathbb R))$, Lebesgue measure is the unique measure $\lambda$ on $\mathcal B(\mathbb R)$ which assigns to intervals $$\lambda(a, b) = \lambda[a, b) = \lambda(a, b] = \lambda[a,b] = b – a.$$ That is, Lebesgue measure assigns to Borel sets what me regard as their usual length/size.

Exercise: Show that Lebesgue measure on $[0,1]$ gives a probability measure on the Borel $\sigma$-algebra $\mathcal B[0,1] = \{ B \cap [0,1] : B \in \mathcal B(\mathbb R)\}$.

Like in the previous exercise, we can make new measure spaces from old by restricting a measure to a given set in its $\sigma$-algebra. Specifically, if $A \in \mathcal E$, then $\mathcal E_A = \{E \cap A : E \in \mathcal E\}$ is a $\sigma$-algebra on $A$ and $(A, \mathcal E_A, \mu)$ is a measure space called the trace of $(E, \mathcal E, \lambda)$ on $A$. Given a probability space $(\Omega, \mathcal H, \mathbb P)$ and an event $A \in \mathcal H$ with $\mathbb P(A) > 0$, then $(A, \mathcal H_A, \mathbb P(\cdot)/\mathbb P(A) )$ is a probability space.

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