measure space

A measure space is a triple \[(X,\mathcal{F},\mu)\] where \[X\] is a set, \[\mathcal{F}\] is a sigma-algebra on the set \[X\] and \[\mu\] is a measure on \[(X,\mathcal{F})\]. Additionally, \[\mu\] must satisfy the property: if \[\left( A_{n} \right)^{\infty}_{n=1}\] are pair-wise disjoint then \[\mu(\cup_{n=1}^{\infty}A_{n})=\sum_{n=1}^{\infty}\mu(A_{n})\].

\[X\] is the place that we want to "measure" parts of. It could be \[\mathbb{R}^{n}\] or some possible outcomes of an experiment like coin tossing. A sigma-algebra, despite having a menacing name, is just a set of subsets of \[X\] that we will be "measuring".

To make this more understandable, assume we're going to model a coin flip. Set \[X=\left\{ -1,1 \right\}\], where \[-1\] corresponds to tails and \[1\] corresponds to head (it can also be any number of our liking). Set \[\mathcal{F}=\mathcal{P}(X)\], where \[\mathcal{P}(X)\] stands for the power set of \[X\]. In this simple case, \[\mathcal{F}\] consists of \[\mathcal{F}=\mathcal{P}(X)=\left\{ \emptyset, \left\{ -1 \right\},\left\{ 1 \right\},\left\{ -1,1 \right\} \right\}\]. As for the measure, statistically we're equally likely to get head or tails, so \[\mu(\left\{ -1 \right\})=\mu(\left\{ 1 \right\})=\frac{1}{2}\]. This is also known as a probability space.