By Elias M. Stein
Real Analysis is the 3rd quantity within the Princeton Lectures in research, a sequence of 4 textbooks that goal to offer, in an built-in demeanour, the middle components of study. right here the point of interest is at the improvement of degree and integration concept, differentiation and integration, Hilbert areas, and Hausdorff degree and fractals. This ebook displays the target of the sequence as a complete: to make simple the natural solidarity that exists among a number of the components of the topic, and to demonstrate the broad applicability of rules of study to different fields of arithmetic and science.
After surroundings forth the elemental evidence of degree concept, Lebesgue integration, and differentiation on Euclidian areas, the authors movement to the weather of Hilbert area, through the L2 conception. They subsequent current uncomplicated illustrations of those options from Fourier research, partial differential equations, and complicated research. the ultimate a part of the e-book introduces the reader to the interesting topic of fractional-dimensional units, together with Hausdorff degree, self-replicating units, space-filling curves, and Besicovitch units. each one bankruptcy has a chain of routines, from the rather effortless to the extra complicated, which are tied on to the textual content. a considerable variety of tricks inspire the reader to tackle even the tougher exercises.
As with the opposite volumes within the sequence, Real Analysis is on the market to scholars attracted to such diversified disciplines as arithmetic, physics, engineering, and finance, at either the undergraduate and graduate levels.
Also on hand, the 1st volumes within the Princeton Lectures in Analysis:
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Additional info for Real analysis : measure theory, integration, and Hilbert spaces
Sample text
3 Suppose f is measurable on Rd . Then there exists a sequence of step functions {ψk }∞ k=1 that converges pointwise to f (x) for almost every x. Proof. By the previous result, it suffices to show that if E is a measurable set with finite measure, then f = χE can be approximated by step functions. 4, which states that for every there exist cubes Q1 , . . , QN such that N m(E j=1 Qj ) ≤ . By considering the grid formed by extending the sides of these cubes, we see that there exist almost disjoint rectangles ˜ ˜1, .
Then, FN (x) → f (x) as N tends to infinity for all x. Now, we partition the range of FN , namely [0, N ], as follows. For fixed N, M ≥ 1, we define E ,M = x ∈ QN : M < FN (x) ≤ +1 M , for 0 ≤ < N M . Then we may form FN,M (x) = M χE ,M (x). Each FN,M is a simple function that satisfies 0 ≤ FN (x) − FN,M (x) ≤ 1/M for all x. If we now choose N = M = 2k with k ≥ 1 integral, and let ϕk = F2k ,2k , then we see that 0 ≤ FM (x) − ϕk (x) ≤ 1/2k for all x, {ϕk } is increasing, and this sequence satisfies all the desired properties.
Similar conclusions hold for whichever combination of strict or weak inequalities one chooses. For example, if f is finite-valued, then it is measurable if and only if {a ≤ f < b} for all a, b ∈ R. By the same arguments one sees the following: Property 1 The finite-valued function f is measurable if and only if f −1 (O) is measurable for every open set O, and if and only if f −1 (F ) is measurable for every closed set F . Note that this property also applies to extended-valued functions, if we make the additional hypothesis that both f −1 (∞) and f −1 (−∞) are measurable sets.