1.3. The Lebesgue integral 47 integrate functions f that are not defined everywhere on Rd, but merely de- fined almost everywhere on Rd (i.e. f is defined on some set Rd\N where N is a null set), simply by extending f to all of Rd in some arbitrary fashion (e.g. by setting f equal to zero on N). This is extremely convenient for analysis, as there are many natural functions (e.g. sin x x in one dimension, or 1 |x|α for various α 0 in higher dimensions) that are only defined almost everywhere instead of everywhere (often due to “division by zero” problems when a denominator vanishes). While such functions cannot be evaluated at certain singular points, they can still be integrated (provided they obey some integrability condition, of course, such as absolute integrability), and so one can still perform a large portion of analysis on such functions. In fact, in the subfield of analysis known as functional analysis, it is convenient to abstract the notion of an almost everywhere defined function somewhat, by replacing any such function f with the equivalence class of almost everywhere defined functions that are equal to f almost everywhere. Such classes are then no longer functions in the standard set-theoretic sense (they do not map each point in the domain to a unique point in the range, since points in Rd have measure zero), but the properties of various func- tion spaces improve when one does this (various semi-norms become norms, various topologies become Hausdorff, and so forth). See §1.3 of An epsilon of room, Vol. I for further discussion. Remark 1.3.7. The “Lebesgue philosophy” that one is willing to lose con- trol on sets of measure zero is a perspective that distinguishes Lebesgue-type analysis from other types of analysis, most notably that of descriptive set theory, which is also interested in studying subsets of Rd, but can give com- pletely different structural classifications to a pair of sets that agree almost everywhere. This loss of control on null sets is the price one has to pay for gaining access to the powerful tool of the Lebesgue integral if one needs to control a function at absolutely every point, and not just almost every point, then one often needs to use tools other than integration theory (unless one has some regularity on the function, such as continuity, that lets one pass from almost everywhere true statements to everywhere true statements). 1.3.2. Measurable functions. Much as the piecewise constant integral can be completed to the Riemann integral, the unsigned simple integral can be completed to the unsigned Lebesgue integral, by extending the class of unsigned simple functions to the larger class of unsigned Lebesgue measur- able functions. One of the shortest ways to define this class is as follows: Definition 1.3.8 (Unsigned measurable function). An unsigned function f : Rd → [0, +∞] is unsigned Lebesgue measurable, or measurable for short, if it is the pointwise limit of unsigned simple functions, i.e., if there exists

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