## A Course on Integration Theory: including more than 150 by Nicolas Lerner

By Nicolas Lerner

This textbook presents a close therapy of summary integration conception, development of the Lebesgue degree through the Riesz-Markov Theorem and in addition through the Carathéodory Theorem. it is usually a few simple houses of Hausdorff measures in addition to the elemental homes of areas of integrable services and conventional theorems on integrals reckoning on a parameter. Integration on a product area, swap of variables formulation in addition to the development and learn of classical Cantor units are taken care of intimately. Classical convolution inequalities, reminiscent of Young's inequality and Hardy-Littlewood-Sobolev inequality are confirmed. The Radon-Nikodym theorem, notions of harmonic research, classical inequalities and interpolation theorems, together with Marcinkiewicz's theorem, the definition of Lebesgue issues and Lebesgue differentiation theorem are additional themes incorporated. a close appendix offers the reader with a number of parts of easy arithmetic, corresponding to a dialogue round the calculation of antiderivatives or the Gamma functionality. The appendix additionally presents extra complicated fabric similar to a few simple homes of cardinals and ordinals that are precious within the research of measurability.

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**Example text**

4, we obtain (3). To prove (4), we deﬁne E = {f = +∞}, and we note that μ(E) > 0 implies for all integers n ≥ 1, that f dμ ≥ X entailing X f dμ ≥ n E dμ = nμ(E), E f dμ = +∞. 2. Let (X, M, μ) be a measure space where μ is a positive measure. The space L1 (μ) is deﬁned as the quotient of L1 (μ) (cf. e. (f ∼ g means μ({x ∈ X, f (x) = g(x)}) = 0). 3. e. since μ(N1 ∪ N2 ) = 0. 36 Chapter 1. 1(1). , then X gdμ. 1. 4. Let (X, M, μ) be a measure space where μ is a positive measure. (1) The mapping from L1 (μ) into C deﬁned by f → X f dμ is a linear form.

Then lim inf xn is the smallest accumulation point of the sequence and lim sup xn the largest. We have lim inf xn ≤ lim sup xn and equality holds if and only if the sequence is converging to this value. Proof. Using the homeomorphism ψ0 deﬁned above (cf. 19)) we can assume that (xn )n∈N is a sequence in [−1, 1]. , a limit point of subsequence, (xnk )k∈N , (n0 < n1 < n2 < · · · < nk < nk+1 < · · · ), then y ←− xnk ≤ sup xl −→ lim sup xn , k→+∞ l≥nk k→+∞ where the second limit comes from a subsequence of a converging sequence.

8. Let (X, M, μ) be a measure space where μ is a positive measure. Let f be in L1 (μ) and let (fn )n∈N be a sequence of functions in L1 (μ) such that the following properties hold. , (2) limn fn L1 (μ) = f L1(μ) . Then limn fn − f L1 (μ) = 0. 9. To sum-up, for a sequence (fn ) in L1 (μ), f ∈ L1 (μ), pointwise / fn convergence ⎫ ⎪ ⎬ f ⎪ ⎭ and limn fn L1 (μ) = f =⇒ fn L1 (μ) / f . 6) L1 (μ) The following proposition is an important consequence of the Lebesgue dominated convergence theorem. 10. Let (X, M, μ) be a measure space where μ is a positive measure.