# Lectures on Measure and Integration by Harold Widom

By Harold Widom

Cover

Lectures on degree AND INTEGRATION

INTRODUCTION

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

Back disguise

Best analysis books

Analisi matematica

Nel quantity vengono trattati in modo rigoroso gli argomenti che fanno parte tradizionalmente dei corsi di Analisi matematica I: numeri reali, numeri complessi, limiti, continuità, calcolo differenziale in una variabile e calcolo integrale secondo Riemann in una variabile. Le nozioni di limite e continuità sono ambientate negli spazi metrici, di cui viene presentata una trattazione elementare ma precisa.

Multicriteria and Multiobjective Models for Risk, Reliability and Maintenance Decision Analysis

This booklet integrates a number of standards recommendations and techniques for difficulties in the hazard, Reliability and upkeep (RRM) context. The innovations and foundations relating to RRM are thought of for this integration with multicriteria ways. within the booklet, a basic framework for construction determination versions is gifted and this can be illustrated in a variety of chapters by way of discussing many various determination versions on the topic of the RRM context.

Extremal Lengths and Closed Extensions of Partial Differential Operators

Test of print of Fuglede's paper on "small" households of measures. A strengthening of Riesz's theorem on subsequence is acquired for convergence within the suggest. This result's utilized to calculus of homologies and classes of differential kinds.

Additional info for Lectures on Measure and Integration

Example text

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.