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Showing posts from October, 2022

Measure theory and $L^p$ Spaces

I'm on a quest to learn about operator algebras in the hopes of understanding the many interesting ways they have been applied to quantum field theory. This note is the first in a series that will build the essential aspects of the theory from the ground up. I will not prove everything, giving references for the proofs of many lemmas, but I will try to give enough detail that the mathematical underpinnings of the theory are clear. This post is about $L^p$ spaces. These are Banach spaces of functions — with the $L^{\infty}$ space actually being a Banach algebra — that show up frequently in the theory of general operator algebras. I will introduce the basics of measure theory and the general theory of Lebesgue integration, introduce the $L^p$ spaces, and prove some basic theorems concerning them. The outline is: In section 1 , I will introduce the basic tools of measure theory: $\sigma$-algebras, measurable functions, and measures. In section 2 , I will discuss some elementary proper