Part III (vi) Continuing on, there is the subject of measure theory. That was from H. Lebesgue, a student of E. Borel, right, in France near 1900. They were

Part III (vi) Continuing on, there is the subject of measure theory. That was from H. Lebesgue, a student of E. Borel, right, in France near 1900. They were correct: They improved on the Riemann integral. Don't worry: Whenever the Riemann integral exists, the integral from Lebesgue's measure theory gives the same numerical answer. So, why a new (way of defining the) integral? Two biggie reasons: (a) For a lot of theorem proving about integrals, e.g., differentiation under the integral sign, clean treatment of what physics needed from the Dirac delta function (right, there can be no such function, but measure theory has a good answer), definition of an integral of several variables, interchange of order of integration in iterated integrals, tying off some old, loose ends in Fourier theory, the deep connections between integration and linear operators, and more, Lebesgue's work is crucial and terrific stuff. (b) Lebesgue's integral is much more general than the Riemann integral, and that generality is crucial, especially as the foundation for probability theory. Now, for what Lebesgue did: First, he developed measure theory. That's essentially just a grown up theory of area like you have known about since grade school. E.g., given a set, in the real numbers, R^n, something more complicated, or something fully abstract, the measure of that set is essentially just its area. With the generalization, sure, can have some sets with measure infinity, negative measure, complex valued measure, etc. But on the real line, with the usual measure, Lebesgue measure, the measure of an interval is just its length, and you already know about that. But Lebesgue measure is darned general: It's tricky to show that there is a set of the reals that is not Lebesgue measurable, and the usual proof uses the axiom of choice. So, for measure theory, there is a measure space with three things: There is a space, just some non-empty set, say, M; there is a collection of measurable sets, subsets of M called, say, S; and there is a measure m. Then the collection of measurable sets S satisfies the simple, essentially obvious axioms we would want for area and, thus, is a sigma algebra of sets. Then for each set A in S, its measure is the real (or complex) number m(A). We ask that m have the properties we want for a good theory of area. It's just a grown up version of area. The theory is not trivial; it was tricky to get all the details just right so that have a good theory of area, that is, so that area works like we want it to. E.g., for the space, can have the set of real numbers R. For the measurable sets, have the intervals and, then, all the other sets needed to have a sigma algebra (a short proof shows that this definition is well defined). Then the Lebesgue measure of an interval is just its length, and the measure of all the other measurable sets are what they, then, have to be (need some theorems and proofs here). So, for probability theory, a probability space is just a measure space (each point in that space is one experimental trial); a probability, P, is just a positive measure with maximum value 1; an event A is just one of the measurable sets; and the probability of A is just its measure P(A). What we want for probability is already so close to a theory of area that, really, we have little choice but just to follow what Lebesgue did. That's what A. Kolmogorov observed in his 1933 paper. Second, with the foundation of measure theory, Lebesgue defined the Lebesgue integral. So, what is being integrated is (usually) a function taking real or complex values. The domain of the function is a measure space. Then for the integral, say, in the case of a real valued function, we partition on the Y axis, that is, in the range of the function instead of in its domain. So, we don't have to do Riemann-like partitions of the domain of the function and, thus, the domain can be much more general. As the first step, we only integrate functions that are >= 0, and we do that by approximating, right, again with essentially rectangles, only from below the function, not both above and below as for Riemann. Here the domain of the function can be the whole space, e.g., the whole real line. We don't care about either continuity or compactness. For a function that is both positive and negative, we multiply the negative part by -1 and integrate the two parts separately. If at least one of the two results is not infinity, then we subtract, and that's the integral. A random variable is just such a function, and its expectation is just its integral. Summary Don't feel like the Lone Ranger; not everyone knows this stuff. E.g., from all I've been able to see from quantum mechanics, the wave functions are differentiable. Then I'm told that the wave functions, wondrous, are also continuous. From baby Rudin, of course they are continuous; every differentiable function is continuous! Then I'm told that the wave functions form a Hilbert space. Well, I can see that they can be points in a Hilbert space, but they can't form a Hilbert space because the continuous functions won't be complete. In elementary probability, it is common, e.g., for finding the expectation of a Gaussian random variable, to integrate over the whole real line. Tilt: Baby Rudin defines the Riemann integral only over compact sets. So, can use an improper integral, and then they want to differentiate under the integral sign. Tilt again -- the needed theorems are not so easy to find. So, really, this common stuff, in probability, physics, etc. of integrating over the whole real line or over all of R^n is using the Lebesgue theory where have clean theorems for such things. That's your overview and your work for the weekend. You are now permitted two beers and half a pizza but only if you have someone you like a lot for the rest of the pizza and some more beer. Can substitute Chianti for the beer. But no more math permitted; maybe a movie, but not math!