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The Banach Algebra of Borel Measures on Euclidean Space

This blog post is intended to deliver a quick explanation of the algebra of Borel measures on $\mathbb{R}^n$. It will be broken into pieces. All complex-valued complex Borel measures $M(\mathbb{R}^n)$ clearly form a vector space over $\mathbb{C}$. The main goal of this post is to show that this is a Banach space and also a Banach algebra.

In fact, the $\mathbb{R}^n$ case can be generalised into any locally compact abelian group (see any abstract harmonic analysis books), this is because what really matters here is being locally compact and abelian. But at this moment we stick to Euclidean spaces. Note since $\mathbb{R}^n$ is $\sigma$-compact, all Borel measures are regular.

To read this post you need to be familiar with some basic properties of Banach algebra, complex Borel measures, and the most important, Fubini’s theorem.

The Banach Algebra of Borel Measures on Euclidean Space

Quasi-analytic Classes

We study the concept of quasi-analytic functions, which are quite close to being analytic.
Quasi-analytic Classes

Several ways to prove Hardy's inequality

Several ways to prove Hardy's inequality

A Continuous Function Sending L^p Functions to L^1

A Continuous Function Sending L^p Functions to L^1

Basic Facts of Semicontinuous Functions

Basic Facts of Semicontinuous Functions

More properties of zeros of an entire function

More properties of zeros of an entire function

The Lebesgue-Radon-Nikodym theorem and how von Neumann proved it

The Lebesgue-Radon-Nikodym theorem and how von Neumann proved it

Topological properties of the zeros of a holomorphic function

Topological properties of the zeros of a holomorphic function