148 1. Measure theory (v) Explain why the above results give an alternate proof of Exercise 1.6.4 and of Proposition 1.6.41. Remark 1.6.43. As the above exercise indicates, the Henstock-Kurzweil integral (also known as the Denjoy integral or Perron integral) extends the Riemann integral and the absolutely convergent Lebesgue integral, at least as long as one restricts attention to functions that are defined and are finite everywhere (in contrast to the Lebesgue integral, which is willing to tolerate functions being infinite or undefined so long as this only occurs on a null set). It is the notion of integration that is most naturally associated with the fundamental theorem of calculus for everywhere differentiable functions, as seen in part (iv) of the above exercise it can also be used as a unified framework for all the proofs in this section that invoked Cousin’s theorem. The Henstock-Kurzweil integral can also integrate some (highly oscillatory) functions that the Lebesgue integral cannot, such as the derivative F of the function F appearing in Exercise 1.6.53. This is analogous to how condi- tional summation limN→∞ ∑N n=1 an can sum conditionally convergent series ∑∞ n=1 an, even if they are not absolutely integrable. However, in as much as conditional summation is not always well behaved with respect to rearrange- ment, the Henstock-Kurzweil integral does not always react well to changes of variable also, due to its reliance on the order structure of the real line R, it is diﬃcult to extend the Henstock-Kurzweil integral to more general spaces, such as the Euclidean space Rd, or to abstract measure spaces. 1.7. Outer measures, pre-measures, and product measures In this text so far, we have focused primarily on one specific example of a countably additive measure, namely Lebesgue measure. This measure was constructed from a more primitive concept of Lebesgue outer measure, which in turn was constructed from the even more primitive concept of elementary measure. It turns out that both of these constructions can be abstracted. In this section, we will give the Carath´ eodory extension theorem, which constructs a countably additive measure from any abstract outer measure this gener- alises the construction of Lebesgue measure from Lebesgue outer measure. One can in turn construct outer measures from another concept known as a pre-measure, of which elementary measure is a typical example. With these tools, one can start constructing many more measures, such as Lebesgue-Stieltjes measures, product measures, and Hausdorff measures. With a little more effort, one can also establish the Kolmogorov extension theorem, which allows one to construct a variety of measures on infinite- dimensional spaces, and is of particular importance in the foundations of probability theory, as it allows one to set up probability spaces associated

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