---
tag: N4KDT83V3S
kind: concept
name: Measurable space
msc: [28A05]
sources:
  - url: https://en.wikipedia.org/wiki/Measurable_space
same_as:
  wikidata: Q3058218
  wikipedia: Measurable space
---

## Summary

A measurable space is a set together with a [[σ-algebra|0SV8K0TDGY]] of its subsets.
The sets in the σ-algebra are the measurable sets.
Measurable spaces are the setting in which measures, measurable functions and random variables are defined.

## Explanations

### L1 human

A measurable space is a set with a chosen list of subsets that one is allowed to measure.
The list must contain the whole set and be stable under complements and countable unions.
Think of a dartboard: one wants to speak of the probability of hitting a region, so one fixes which regions count.
For example, the real line with the Borel sets, generated by the open intervals, is a measurable space.
No measure is chosen yet; the measurable space only fixes which sets will get a size.