Measure space

 ( Measure Theory ) 

• $X$ is a set
• $\backslash mathcal\; A$ is a -algebra on the set $X$
• $\backslash mu$ is a measure on $(X,\; \backslash mathcal\{A\})$
• $\backslash mu$ must satisfy countable additivity. That is, if $(A\_\{n\})\_\{n=1\}^\{\backslash infty\}$ are pair-wise disjoint then $\backslash mu(\backslash cup\_\{n=1\}^\{\backslash infty\}A\_\{n\})\; =\backslash sum\_\{n=1\}^\{\backslash infty\}\backslash mu(A\_\{n\})$

In other words, a measure space consists of a measurable space $(X,\; \backslash mathcal\{A\})$ together with a measure on it.

In this simple case, the power set can be written down explicitly: $$\backslash wp(X)\; =\; \backslash \{\backslash varnothing,\; \backslash \{0\backslash \},\; \backslash \{1\backslash \},\; \backslash \{0,\; 1\backslash \}\backslash \}.$$

As the measure, define $\backslash mu$ by $$\backslash mu(\backslash \{0\backslash \})\; =\; \backslash mu(\backslash \{1\backslash \})\; =\; \backslash frac\{1\}\{2\},$$ so $\backslash mu(X)\; =\; 1$ (by additivity of measures) and $\backslash mu(\backslash varnothing)\; =\; 0$ (by definition of measures).

This leads to the measure space $(X,\; \backslash wp(X),\; \backslash mu).$ It is a probability space, since $\backslash mu(X)\; =\; 1.$ The measure $\backslash mu$ corresponds to the Bernoulli distribution with $p\; =\; \backslash frac\{1\}\{2\},$ which is for example used to model a fair coin flip.

• Probability spaces, a measure space where the measure is a probability measure
• Finite measure spaces, where the measure is a finite measure
• $\backslash sigma$-finite measure spaces, where the measure is a $\backslash sigma$-finite measure

Another class of measure spaces are the complete measure spaces.