Last time I introduced the notion of the weak topology on a Banach space . I defined the topology in terms of net convergence, but remarked that this was equivalent to the weak topology generated by . The purpose of … Continue reading
Well, it’s been a long time since I’ve written a post, but I have a bit of time tonight and I really don’t feel like doing homework tonight, so let’s talk about some mathematics! Last time I wrote, I promised … Continue reading
Recall that in the finite dimensional world, two vector spaces are isometrically isomorphic if and only if they have the same dimension. In the last post, I mentioned that, by using the appropriate definition of a basis and dimension, one … Continue reading
So far we’ve proven Littlewood’s first and third principles. In this last installment, we’ll prove his second principle, that “every function is nearly continuous.” This result is better known as Lusin’s Theorem.
Littlewood’s three principles provide useful intuition for those first learning measure theory: “The extent of knowledge [of real analysis] required is nothing like as great as is sometimes supposed. There are three principles, roughly expressible in the following terms: Every … Continue reading
When I first studied measure theory, I felt like I could never quite wrap my head around what a measurable set or function looked like. After all, the Borel Heirarchy is huge, and what the hell does a set look … Continue reading
Last time I mentioned the paradoxical-sounding situation in which I said “pick a number at random under some given continuous probability distribution over the reals, and call this number ”. But what was the probability of choosing ? Well, a … Continue reading