L1 Space

Introduction

Let $(X,{\mathcal {M}},\mu )$ be a measure space. From our study of integration, we know that if $f,g$ are integrable functions, the following functions are also integrable:

1. $f+g$
2. $cf$, for $c\in \mathbb {R}$

This shows that the set of integrable functions on any measurable space is a vector space. Furthermore, integration is a linear functional on this vector space, ie a linear function sending elements in our vector space to $\mathbb {R}$, one would like to use integration to define a norm on our vector space. However, if one were to check the axioms for a norm, one finds that if $f=0$ almost everywhere, then $\int f=\int 0=0$. This motivates our definition of $L^{1}(\mu )$ to be the set of integrable functions up to equivalence to sets of measure zero.

$L^{1}(\mu )$ Space

In this section, we will construct $L^{1}(\mu )$.

Definition: Let $L^{1}$ denote the set of integrable functions on $X$, ie $\int |f|<\infty$. Define an equivalence relation: $f\sim g$ if $f=g$ a.e. Then Failed to parse (unknown function "\coloneqq"): {\displaystyle L^1(\mu)\coloneqq L^1/\sim} .