# A Primer of Lebesgue Integration by H. S. Bear By H. S. Bear

The Lebesgue imperative is now usual for either functions and complicated arithmetic. This books begins with a assessment of the commonplace calculus essential after which constructs the Lebesgue imperative from the floor up utilizing an identical principles. A Primer of Lebesgue Integration has been used effectively either within the lecture room and for person study.

Bear provides a transparent and easy advent for these cause on extra examine in better arithmetic. also, this publication serves as a refresher supplying new perception for these within the box. the writer writes with a fascinating, common-sense variety that appeals to readers in any respect degrees.

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For x e S - FN? I fki^) - f{x)\ < ^ foi" all k>N. Illll Problem 5. e. the same result holds. ""HI The next proposition gives the form in which it is easiest to remember and apply the above result. Proposition 7. (Egoroffs Theorem) If {fn} is a sequence of measurable functions on a finite measure set Sy and fn —> f pointwise on S, then for every 8 > 0 there is a measurable set E c S of measure less than 8 so that fn —> f uniformly on S — E. Proof For each n we find a set E„ of measure less than 8/2^ and a number N„ so that \fk{x) — f{x)\ < ^ior k> Nn and x e S — En.

If f is a hounded function which is integrable on a set S of finite measure, then f is the ax. pointwise limit of simple functions, and hence f is measurable. Proof. For each n we let Pn be a partition of S such that U( f, Pn) - L( /", Pn) < 1/n. We can assume that Pi>P2> P3 >- • • • by replacing each P„ by the common refinement of its predecessors. Let P„ = {Eni} and mni = 'mi{f(x) : x e Eni} Mm =sup{/"(x) : xe Eni). Let (pn be the simple function which has the value m^ on E„/, and let V^ be M^ on Eni • Then \jfn — ^« is a simple function and j {fn - (pn) = Yl^Mni = " mni)fl(Eni) U(fPn)-L(fPn).

If Sis a set of finite measure and m < f(x) < M for all X e S, and P, Qare partitions of S with Q>Py then mniS) < L(f P) < L(f Q) < U(f Q) < U(f P) < M/x(5). Proof Let P = {£,} and Q = {Fij} with U/ P// = £/ for each /. Let nii = inf {/"(x) : x e £/} Tftij = inf { /"(x) : X e Fij}, Then clearly mi < mij for all /, /, so L(/-,P) = ^ m , / x ( £ , )