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 integration fails to be a norm by taking $f=0$ almost everywhere, then $\int f=\int 0=0$. In other words, there are non zero functions which has a zero integral. 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 $L^{1}(\mu )=L^{1}/\sim$.

To make sense of the definition, we need the following proposition:

Proposition: Let $f,g\in L^{1}$, then the following are equivalent:

1. $\int _{E}f=\int _{E}g,$ for all $E\in {\mathcal {M}}$
2. $\int |f-g|=0$
3. $f=g$ a.e.

Proof

$2\implies 1:{\bigg \vert }\int _{E}f-\int _{E}g{\bigg \vert }\leq 1_{E}|f-g|\leq \int |f-g|=0$

$3\implies 2:$ Since $f=g$ a.e., $|f-g|=0$ a.e. Take a simple function, $\phi$, such that $0\leq \phi \leq |f-g|$, such $\phi$ must be $0$ a.e. Therefore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int |f-g|=\sup\left\{\int \phi: 0\leq\phi\leq|f-g|,\phi\text{ simple}\right\}=0}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1\implies 3: } Suppose the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ x\in X: f(x)\neq g(x)\} } does not have measure zero. Then either Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_+=\{x\in X: (f-g)_+(x)\neq 0\} } or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_-=\{x\in X: (f-g)_-(x)\neq0\}} has nonzero measure, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f-g)_+} denotes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \max\{f-g,0\}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f-g)_-} denotes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \max\{-(f-g),0\}} . WLOG, assume Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_+} has nonzero measure. Define the following sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{+,n}=\{x\in X: (f-g)_+>1/n\}} , then from continuity from below, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu(E_+)=\mu\left(\cup_i^\infty E_{+,i}\right)=\lim_{i\to\infty}\mu(E_{+,i})>0} . This shows that there exists some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\in\mathbb{N}} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu(E_{+,i})>0} , which implies that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_E (f-g)\geq\int_E(f-g)_+\geq \int_{E_{+,i}}(f-g)_+\geq \mu(E_{+,i})/i>0} , contradicting 1.

With the proposition, we define our norm on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^1(\mu)} to be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lVert f\rVert=\int |f|} . This is indeed a norm since:

1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int |f+g| \leq \int |f|+\int|g|}
2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int |cf|=c\int |c||f|, c\in \mathbb{R}}
3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int |f|=0\iff f=0} a.e