Measure space

Remarks

Let $X = \set{1, \ldots, 6}$, let $\Sigma$ be the power set of $X$, and let $\mu(A) = \abs{A} / 6$. Then $\Sigma$ is a σ-algebra and $\mu$ is a measure, since the number of elements is additive over disjoint sets, so $(X, \Sigma, \mu)$ is a measure space. On $\N$, let $\mathcal{A}$ be the family of finite sets and of sets with finite complement, and let $\mu(A) = \abs{A}$ for finite $A$ and $\mu(A) = \infty$ otherwise. Then $(\N, \mathcal{A}, \mu)$ is not a measure space, since $\mathcal{A}$ is not a σ-algebra: it does not contain the set of even numbers. A measure space with $\mu(X) = 1$ is a probability space.