HOME

TheInfoList



OR:

In mathematics, Scheffé's lemma is a proposition in
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
concerning the
convergence Convergence may refer to: Arts and media Literature *''Convergence'' (book series), edited by Ruth Nanda Anshen *Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics: **A four-part crossover storyline that ...
of sequences of integrable functions. It states that, if f_n is a sequence of integrable functions on a
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
(X,\Sigma,\mu) that converges almost everywhere to another integrable function f, then \int , f_n - f, \, d\mu \to 0 if and only if \int , f_n , \, d\mu \to \int , f , \, d\mu. The proof is based fundamentally on an application of the triangle inequality and
Fatou's lemma In mathematics, Fatou's lemma establishes an inequality (mathematics), inequality relating the Lebesgue integral of the limit superior and limit inferior, limit inferior of a sequence of function (mathematics), functions to the limit inferior of ...
.


Applications

Applied to
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, Scheffe's theorem, in the form stated here, implies that almost everywhere pointwise convergence of the
probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
s of a sequence of \mu-
absolutely continuous In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship betwe ...
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s implies
convergence in distribution In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
of those random variables.


History

Henry Scheffé published a proof of the statement on convergence of probability densities in 1947. The result is a special case of a theorem by
Frigyes Riesz Frigyes Riesz (, , sometimes known in English and French as Frederic Riesz; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> ...
about convergence in L''p'' spaces published in 1928.


References

{{DEFAULTSORT:Scheffe's lemma Theorems in measure theory