You are here
Homeinfinite product measure
Primary tabs
infinite product measure
Let $(E_{i},\mathcal{B}_{i},\mu_{i})$ be measure spaces, where $i\in I$ an index set, possibly infinite. We define the product of $(E_{i},\mathcal{B}_{i},\mu_{i})$ as follows:
1. let $E=\prod E_{i}$, the Cartesian product of $E_{i}$,
2. let $\mathcal{B}=\sigma((\mathcal{B}_{i})_{{i\in I}})$, the smallest sigma algebra containing subsets of $E$ of the form $\prod B_{i}$ where $B_{i}=E_{i}$ for all but a finite number of $i\in I$.
Then $(E,\mathcal{B})$ is a measurable space. The next task is to define a measure $\mu$ on $(E,\mathcal{B})$ so that $(E,\mathcal{B},\mu)$ becomes in addition a measure space. Before proceeding to define $\mu$, we make the assumption that
each $\mu_{i}$ is a totally finite measure, that is, $\mu_{i}(E_{i})<\infty$.
In fact, we can now turn each $(E_{i},\mathcal{B}_{i},\mu_{i})$ into a probability space by introducing for each $i\in I$ a new measure:
$\overline{\mu}_{i}=\frac{\mu_{i}}{\mu_{i}(E_{i})}.$ 
With the assumption that each $(E_{i},\mathcal{B}_{i},\mu_{i})$ is a probability space, it can be shown that there is a unique measure $\mu$ defined on $\mathcal{B}$ such that, for any $B\in\mathcal{B}$ expressible as a product of $B_{i}\in\mathcal{B}_{i}$ with $B_{i}=E_{i}$ for all $i\in I$ except on a finite subset $J$ of $I$:
$\mu(B)=\prod_{{j\in J}}\mu_{j}(B_{j}).$ 
Then $(E,\mathcal{B},\mu)$ becomes a measure space, and in particular, a probability space. $\mu$ is sometimes written $\prod\mu_{i}$.
Remarks.

If $I$ is infinite, one sees that the total finiteness of $\mu_{i}$ can not be dropped. For example, if $I$ is the set of positive integers, assume $\mu_{1}(E_{1})<\infty$ and $\mu_{2}(E_{2})=\infty$. Then $\mu(B)$ for
$B:=B_{1}\times\prod_{{i>1}}E_{i}=B_{1}\times E_{2}\times\prod_{{i>2}}E_{i}% \mbox{, where }B_{1}\in\mathcal{B}_{1}$ would not be welldefined (on the one hand, it is $\mu_{1}(B_{1})<\infty$, but on the other it is $\mu_{1}(B_{1})\mu_{2}(E_{2})=\infty$).

The above construction agrees with the result when $I$ is finite (see finite product measure).
Mathematics Subject Classification
28A35 no label found60A10 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff