L1 Space

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Let be a measure space. From our study of integration[1], 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.


In this section, we will construct . These are sometimes called Lebesgue spaces.


Let denote the set of integrable functions on , ie . Define an equivalence relation: if a.e. Then . In some abuse of notation, we often refer to an element as a function, even though it really denotes the equivalence class of all functions in which are a.e. equivalent to .

To see that is indeed an equivalence relation, reflexivity and symmetry are immediate. Transitivity when and follows by considering the null set where and differ and similarly for and . Then see that the set where and differ is a subset of the union of the previous two null sets and hence is also a null set, so .

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

Proposition: Let , then the following are equivalent:

  1. for all
  2. a.e.


Since a.e., a.e. Take a simple function, , such that , such must be a.e. Therefore,

Suppose the set does not have measure zero. Then either or has nonzero measure, where denotes and denotes . WLOG, assume has nonzero measure. Define the following sets , then from continuity from below, . This shows that there exists some such that , which implies that , contradicting 1.

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

  1. a.e

Completeness of space

A space with a metric is said to be complete if for every Cauchy sequence in (that is, as ) there exist such that in the sense that


Riesz-Fischer Theorem

The vector space is complete in its metric induced by the -norm. [2]


See Stein and Shakarchi



  1. 1.0 1.1 Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons, ISBN 0471317160, Second edition.
  2. Elias M. Stein and Rami Shakarchi(2005), Real Analysis: measure theory, integration, & hilbert spaces, first edition