---
tag: 4KCBFQS9WR
kind: concept
name: Daniell integral
msc: [28C05, 28A25]
sources:
  - url: https://en.wikipedia.org/wiki/Daniell_integral
same_as:
  wikidata: Q2996956
  wikipedia: Daniell integral
---

## Summary

The Daniell integral is an approach to integration that starts from a positive linear functional on a space of elementary functions.
It does not start from a measure.
The functional must satisfy a continuity condition along decreasing sequences.
It then extends to a large class of integrable functions, and the measure of a set appears afterwards as the integral of its indicator function.
This approach gives the Lebesgue integral and integrals on locally compact spaces in one framework.

## Explanations

### L1 human

The usual route defines lengths or areas first and integrals afterwards.
Daniell's route goes the other way: it starts with an integral that one already knows on easy functions, such as continuous functions or step functions.
The only demand is that the integral behaves well when a sequence of functions shrinks down to zero.
For example, the Riemann integral of continuous functions with compact support on the real line is such a starting point, and its extension is the Lebesgue integral.