Littlewood's three principles of real analysis

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Littlewood's three principles of real analysis are heuristics of J. E. Littlewood to help teach the essentials of measure theory in mathematical analysis:

1. a measurable set is nearly an open set;
2. a measurable function is nearly a continuous function; and
3. a convergent sequence of functions is nearly uniformly convergent.

The third of these alludes to Egorov's theorem. The other two are standard results for subsets of the real line. (1) Given a measurable set T and ε > 0 there is a set of open intervals with union U such that the symmetric difference of T and U has Lebesgue measure less than ε. Similarly, (2) given a measurable real function f and ε > 0, one can choose an open set V of the real line such that f is continuous outside of V, and V has Lebesgue measure less than ε.

[edit] External links

• PDF of Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Elias M. Stein and Rami Shakarchi, Ch.1; see p.33