Search results
Results from the WOW.Com Content Network
Indeed, two Lebesgue-measurable functions may be constructed in such a way as to make their composition non-Lebesgue-measurable. The (pointwise) supremum, infimum, limit superior, and limit inferior of a sequence (viz., countably many) of real-valued measurable functions are all measurable as well. [1] [4]
Let f 1, f 2, . . . be a sequence of extended real-valued measurable functions defined on a measure space (S,Σ,μ). If there exists a non-negative integrable function g on S such that f n ≤ g for all n, then
If X = [a,b] ⊆ R and μ is Lebesgue measure, there are sequences (g n) of step functions and (h n) of continuous functions converging globally in measure to f. If f and f n (n ∈ N) are in L p (μ) for some p > 0 and (f n) converges to f in the p-norm, then (f n) converges to f globally in measure. The converse is false.
Measure and integration (as the English translation of the title reads) is a definitive monograph on integration and measure theory: the treatment of the limiting behavior of the integral of various kind of sequences of measure-related structures (measurable functions, measurable sets, measures and their combinations) is somewhat conclusive.
The problem is a differential equation of the form [()] + = for an unknown function y on an interval [a, b], satisfying general homogeneous Robin boundary conditions {() + ′ ′ = + ′ ′ =. The functions p, q, and w are given in advance, and the problem is to find the function y and constants λ for which the equation has a solution.
In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces.They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz ().
If is a function then will denote its domain and the support of , denoted by (), is defined to be the closure of the set { (): ()} in (). For two functions f , g : U → C {\displaystyle f,g:U\to \mathbb {C} } , the following notation defines a canonical pairing : f , g := ∫ U f ( x ) g ( x ) d x . {\displaystyle \langle f,g ...
For (,) a measurable space, a sequence μ n is said to converge setwise to a limit μ if = ()for every set .. Typical arrow notations are and .. For example, as a consequence of the Riemann–Lebesgue lemma, the sequence μ n of measures on the interval [−1, 1] given by μ n (dx) = (1 + sin(nx))dx converges setwise to Lebesgue measure, but it does not converge in total variation.