measureTheory

measureTheory - Physics 129a Measure Theory 071126 Frank...

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Physics 129a Measure Theory 071126 Frank Porter 1 Introduction The rigorous mathematical underpinning of much of what we have discussed, such as the construction of a proper Hilbert space of functions, is to be found in “measure theory”. We thus reFne the concepts in our note on Hilbert spaces here. It should immediately be stated that the term “measure” refers to the notion of measuring the “size” of a set. This is the subject of measure theory. With measure theory, we will Fnd that we can generalize the Riemann notion of an integral. [This note contains all the essential ideas to complete the development begun in the note on Hilbert spaces for the construction of a suitable Hilbert space for quantum mechanics. However, this note is still under consruction, as there remain gaps in the presentation.] Let us motivate the discussion by considering the space, C 2 ( 1 , 1) of complex-valued continuous functions on [ 1 , 1]. We deFne the norm, for any f ( x ) C 2 [ 1 , 1]: || f || 2 = Z 1 1 | f ( x ) | 2 dx. (1) Consider the following sequence of functions, f 1 ,f 2 ,... ,in C 2 [ 1 , 1]: f n ( x )= 1 1 x ≤− 1 /n , nt 1 /n x 1 /n , 11 /n x 1. (2) ±ig. 1 illustrates the Frst few of these functions. This set of functions deFnes a Cauchy sequence, with convergence to the discontinuous function f ( x ± 1 1 x 0, 10 <x 1. (3) Since f ( x ) / C 2 [ 1 , 1], this space is not complete. How can we “complete” such a space? We need to add discontinuous functions somehow. Can we simply add all piecewsie continuous functions? Consider the sequence of piecewise continuous functions, g 1 ,g 2 : g n ( x 0 1 x< 1 /n , 1 1 /n x 1 /n , 01 /n < x 1. (4) 1
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-1 1 -1 1 x Figure 1: A Cauchy sequence of functions in C 2 [ 1 , 1]. This sequence is illustrated in Fig. 1. This sequence converges to the function equa lto1a t x = 0, and zero everywhere else. This isn’t exactly what we have in mind when we say “piecewise discontinuous”, but perhaps it is all right. However, the sequence also converges to f ( x ) = 0. This gives us two functions in our vector space with norm zero, which is not allowed. We need to think some more if we are going to lay a rigorous foundation. Measure theory will permit us to deal with these issues. -1 1 x 0 . . . . 1 1 1 1 1 Figure 2: A Cauchy sequence of piecewise discontinuous functions. The coor- dinate system is moved for each function in order to separate them vertically. 2
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2 The Jordan Measure Consider the set, R n ,o f n -tuples of real numbers. We may deFne a gen- eralization of the closed interval on the real numbers as the “generalized interval”: I ( a, b ) ≡{ x : x ∈R n , with components a i x i b i ,i =1 , 2 ,...,n } . (5) We deFne a familiar measure of the “size” of the set I ( a, b ): measure [ I ( a, b )] = n Y i =1 ( b i a i ) . (6) Let us use this idea to deFne the measure of more general subsets of R n .
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measureTheory - Physics 129a Measure Theory 071126 Frank...

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