---
tag: JXFHBY77RN
kind: concept
name: Littlewood's three principles
msc: [28A20, 26A42]
sources:
  - url: https://en.wikipedia.org/wiki/Littlewood%27s_three_principles_of_real_analysis
same_as:
  wikidata: Q6653163
  wikipedia: Littlewood's three principles of real analysis
---

## Summary

Littlewood's three principles are rules of thumb for working with [[Lebesgue measure|2Z2QVWZ262]] on the real line.
Every measurable set is nearly a finite union of intervals, and every [[measurable function|YYNCJSR0PJ]] is nearly continuous.
Every convergent sequence of measurable functions is nearly uniformly convergent.
Each principle has a precise form, and the last two are [[Lusin's theorem|GZMAH9W2SK]] and [[Egorov's theorem|762Y2NYA7W]].
They let one transfer arguments for simple objects to general measurable ones.

## Explanations

### L1 human

Measurable sets and functions can look wild, but up to a small error they behave like familiar ones.
The word "nearly" always means: after throwing away a set of arbitrarily small measure.
For example, the set of rational numbers in $[0, 1]$ looks scattered everywhere, but its measure is $0$, so it is nearly the empty set.
Its indicator function is discontinuous everywhere, yet it is nearly the zero function, which is continuous.