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MIT18_102s09_lec04 - MIT OpenCourseWare http/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu 18.102 Introduction to Functional Analysis Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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18 LECTURE NOTES FOR 18.102, SPRING 2009 Lecture 4. Thursday, 12 Feb I talked about step functions, then the covering lemmas which are the basis of the definition of Lebesgue measure which we will do after the integral then properties of monotone sequences of step functions. To be definite, but brief, by an interval we will mean [ a, b ) an interval closed on the left and open on the right at least for a little while. This is just so the length of the interval, b a, only vanishes when the interval is empty (not true for closed intervals of course) and also so that we can decompose an interval, in this sense, into two disjoint intervals by choosing any interior point: (4.1) [ a, b ) = [ a, t ) [ t, b ) , a < t < b. Now, by a step function (4.2) f : R −→ C (although often we will restrict to functions with real values) we mean a function which vanishes outside a finite union of disjoint ‘intervals’ and is constant on each of them. Thus f ( R ) is finite the function only takes finitely many values and (4.3) f 1 ( c ) is a finite union of disjoint intervals , c = 0 . It is also often convenient to write a step function as a sum N (4.4) f = c i χ [ a i ,b i ) i =1 of multiples of the characteristic functions of our intervals. Note that such a ‘pre- sentation’ is not unique but can be made so by demanding that the intervals be disjoint and ‘maximal’ so f is does not take the same value on two intervals with a common endpoint. Now, a constant multiple of a step function is a step function and so is the sum of two step functions clearly the range is finite. Really this reduces to checking that the difference [ a, b ) \ [ a , b ) and the union of two intervals is always a union of intervals. The absolute value of a step function is also a step function.
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