L1 Space

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Introduction

Let be a measure space. From our study of integration, we know that if are integrable functions, the following functions are also integrable:

  1. , for

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 , 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 almost everywhere, then . In other words, there are non zero functions which has a zero integral. This motivates our definition of to be the set of integrable functions up to equivalence to sets of measure zero.

Space

In this section, we will construct .

Definition

Let denote the set of integrable functions on , ie . Define an equivalence relation: if a.e. Then .

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

Proposition: Let , then the following are equivalent:

  1. for all
  2. a.e.

Proof

(to be filled in)

With the proposition, we define our norm on to be . This is indeed a norm since:

  1. References