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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. Advertisements
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
On this day a few centuries ago, Whateversuitsyourboat’s very own ([Raspy] Ryan) Hotovy was born! Now, a few millennia later, Hotovy is engaged, pursuing his PhD in mathematics at Texas A&M, and presumably getting better at taking cross products! Happy … Continue reading
Let be given by Here’s the problem: For all , does there exist some (dependent on ) for which ? (Of course, denotes iteration of , times.) If this is true, then notice the orbit of will always converge to the … Continue reading
Lomonosov’s Theorem is an amazing theorem which established that if an operator on a Banach space commutes with a nonzero compact operator, then admits a nontrivial hyperinvariant subspace. As I eluded to in a previous post, Compact operators are as … Continue reading
With those pesky definitions out of the way. I will merely state these two lemmas (which are nontrivial theorems in their own right), but the proofs both long approximations that won’t contribute to this post. Lemma 1 (The Schauder Fixed … Continue reading