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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]
When () <, a set of functions (,,) is uniformly integrable if and only if it is bounded in (,,) and has uniformly absolutely continuous integrals. If, in addition, μ {\displaystyle \mu } is atomless, then the uniform integrability is equivalent to the uniform absolute continuity of integrals.
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.
The simplest example of a direct integral are the L 2 spaces associated to a (σ-finite) countably additive measure μ on a measurable space X. Somewhat more generally one can consider a separable Hilbert space H and the space of square-integrable H-valued functions (,).
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 integral of a non-negative general measurable function is then defined as an appropriate supremum of approximations by simple functions, and the integral of a (not necessarily positive) measurable function is the difference of two integrals of non-negative measurable functions. [1]
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.
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