By Harold Widom
Lectures on degree AND INTEGRATION
© 1969 by way of Litton academic Publishing
TABLE OF CONTENTS
CHAPTER 1: MEASURES
CHAPTER II: INTEGRATION
CHAPTER III: THE THEOREMS OF FUBINI
CHAPTER IV: REPRESENTATIONS OF MEASURES
CHAPTER V: THE LEBESGUE SPACES
CHAPTER VI: DIFFERENTIATION
CHAPTER VII: FOURIER SERIES
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Then there exists a sequence If1, I of non-negative simple functions with f1, Proof. -1 f1,(x) xEk,fl(x)÷nxEfl(x) k1 = The f1, are non-negative simple functions, and clearly for each x we have f1,(x) -÷ f(x); in fact if f is bounded, then the convergence is uniform. All we need do now to complete the proof is to show that the f1, are increasing. Let x c S. If f(x) > n+1, then = n+1 > f(x). Now if f(x) < n+ 1 determine the integer 1? by < f( \ = which implies that In case 1? is even ILL < f(x) < — whence 1, =f1,44 (x).
Y: (X,y) e El = IX: (Lt,y) e El Let THEOREM 1. i. = and ii for each X Similarly Y define be u-finite measures, with E e e X, and is a it-measurable for each y £ Y, and E3, e is a v-measurable function of y. Moreover fv(E)dit = Proof. E = U7. clear that First assume that x B1 with IA, x and p are finite. For E c being pairwise disjoint, it is I y) f XAI(X) = whence XEX(Y) = is a v-measurable function of y because it is a finite linear combination of characteristic functions of v-measurable sets.
Let I and g be integrable. Then I + g is also integrable and = /14 + Proof. We first observe that if I> 0, g 0 and 1—g > 0, then by the linearity of the integral for non-negative functions f(I_g)4 + fgdii f(f—g)4 = = ff4- /14 MEASURES 33 For a real-valued function f on S define N(f) = jx e A: 1(x) < 01 and c A: f(x) > P(f) = hr Then we have almost immediately from the definition, ffd,i f = A P(f) Id1z— f N(f) Successively using the additivity of f( ) f we get f f(f+g)4 (f+g)4- P(f+g) A = —(f+g)dg N(f+g) f f N(f) flflf+ g)+ - f f fN(g)(f+g)4+ fl P(f+ g) - N(f) flN(g) f P(f) flN(f+ g) P(f) flP(g) fP(g)(-f-g)4.