Introduction to Holomorphic Functions of Several Variables, by R.C. Gunning

By R.C. Gunning

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Additional resources for Introduction to Holomorphic Functions of Several Variables, Volume III: Homological Theory

Sample text

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 ( £ , )

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