Littlewood's three principles of real analysis
Encyclopedia
Littlewood's three principles of real analysis
are heuristics of J. E. Littlewood to help teach the essentials of measure theory in mathematical analysis
.
as:
The first principle is based on the fact that the inner measure
and outer measure
are equal for measurable sets, the second is based on Lusin's theorem, and the third is based on Egorov's theorem
.
Bressoud,
and Stein & Shakarchi.
Royden gives the bounded convergence theorem as an application of the third principle. The theorem states that if a uniformly bounded sequence of functions converges pointwise, then their integrals converge to the integral of the limit function. If the convergence were uniform this would be a trivial result, and Littlewood's third principle tells us that the convergence is almost uniform, that is, uniform outside of a set of arbitrarily small measure. Because the sequence is bounded, the contribution to the integrals of the small set can be made arbitrarily small, and the integrals on the remainder converge because the functions are uniformly convergent there.
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
are heuristics of J. E. Littlewood to help teach the essentials of measure theory in mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
.
The principles
Littlewood stated the principles in his 1944 Lectures on the Theory of Functionsas:
The first principle is based on the fact that the inner measure
Inner measure
In mathematics, in particular in measure theory, an inner measure is a function on the set of all subsets of members of a given σ-algebra with values in the extended real numbers satisfying some technical conditions...
and outer measure
Outer measure
In mathematics, in particular in measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. A general theory of outer measures was first introduced by...
are equal for measurable sets, the second is based on Lusin's theorem, and the third is based on Egorov's theorem
Egorov's theorem
In measure theory, an area of mathematics, Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions...
.
Example
Littlewood's three principles are quoted in several real analysis texts, for example Royden,Bressoud,
and Stein & Shakarchi.
Royden gives the bounded convergence theorem as an application of the third principle. The theorem states that if a uniformly bounded sequence of functions converges pointwise, then their integrals converge to the integral of the limit function. If the convergence were uniform this would be a trivial result, and Littlewood's third principle tells us that the convergence is almost uniform, that is, uniform outside of a set of arbitrarily small measure. Because the sequence is bounded, the contribution to the integrals of the small set can be made arbitrarily small, and the integrals on the remainder converge because the functions are uniformly convergent there.