Difference between revisions of "Measure space"

Latest revision as of 15:24, 21 July 2015

Note: This page requires knowledge of measurable spaces.

Definition

A measure space^[1] is a tuple:

• $(X,\mathcal{A},\mu:\mathcal{A}\rightarrow[0,+\infty])$ - but because Mathematicians are lazy we simply write:
  • $$(X,\mathcal{A},\mu)$$

Where $X$ is a set, and $\mathcal{A}$ is a $\sigma$-algebra on that set (which together, as $(X,\mathcal{A})$, form a measurable space) and $\mu$ is a measure.

Pre-measure space

Given a set $X$ and an algebra, $\mathcal{A}$ (NOT a $\sigma$-algebra) we can define a pre-measure space^[2] as follows:

• $(X,\mathcal{A},\mu_0)$ where $\mu_0$ is a Pre-measure (a mapping, $\mu_0:\mathcal{A}\rightarrow[0,+\infty]$ with certain properties)

the tuple $(X,\mathcal{A} )$ are a pre-measurable space

See also

• Pre-measurable space
• Measurable space
• Probability space
• Pre-measure
• Measure

References

1. Jump up ↑ Measures, Integrals and Martingales - Rene L. Schilling
2. Jump up ↑ Alec's own terminology. It is likely not in books because it's barely worth a footnote