---
tag: E77T9G4DV7
kind: definition
name: Littlewood's three principles
msc: [28A20, 26A42]
defines: JXFHBY77RN
uses: [2Z2QVWZ262, YYNCJSR0PJ, 0X3WGX95FE, 08VZ57PT26]
sources:
  - url: https://en.wikipedia.org/wiki/Littlewood%27s_three_principles_of_real_analysis
---

## Definition

Let $\lambda$ be [[Lebesgue measure|2Z2QVWZ262]] on $\R$, and let "nearly" mean "outside a set of arbitrarily small measure".
*Littlewood's three principles* are the following three heuristic statements:

1. Every Lebesgue measurable set of finite measure is nearly a finite union of intervals.
2. Every [[measurable function|YYNCJSR0PJ]] is nearly [[continuous|0X3WGX95FE]].
3. Every pointwise convergent sequence of measurable functions is nearly [[uniformly convergent|08VZ57PT26]].

## Remarks

The precise forms on a set $E \subseteq \R$ with $\lambda(E) < \infty$ are as follows.
For principle 1: for every $\varepsilon > 0$ there is a finite union $U$ of intervals with $\lambda(E \mathbin{\triangle} U) < \varepsilon$.
For principle 2: this is [[Lusin's theorem|GZMAH9W2SK]], which gives a closed $F \subseteq E$ with $\lambda(E \setminus F) < \varepsilon$ on which $f$ is continuous.
For principle 3: this is [[Egorov's theorem|762Y2NYA7W]], which gives uniform convergence outside a set of measure less than $\varepsilon$.
Example: for $f = \mathbf{1}_{\Q}$ on $[0, 1]$, take $F$ a closed set of irrationals with $\lambda(F) > 1 - \varepsilon$; then $f|_F = 0$ is continuous.
Without finite measure, principle 3 fails: on $\R$, the functions $\mathbf{1}_{[n, \infty)}$ tend to $0$ pointwise, but not uniformly outside any set of finite measure.

## Explanations

### L1 human

The three principles say that measure theory is close to ordinary analysis, as long as one is allowed to ignore a small set.
Sets are almost intervals, functions are almost continuous, and convergence is almost uniform.