L1 Space: Difference between revisions
Jump to navigation
Jump to search
No edit summary |
No edit summary |
||
| Line 1: | Line 1: | ||
==Introduction== | |||
Let <math>(X,\mathcal{M},\mu)</math> be a measure space. From our study of integration, we know that if <math> f,g</math> are integrable functions, the following functions are also integrable: | Let <math>(X,\mathcal{M},\mu)</math> be a measure space. From our study of integration, we know that if <math> f,g</math> are integrable functions, the following functions are also integrable: | ||
| Line 9: | Line 9: | ||
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 <math>\mathbb{R}</math>, 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 <math>f=0</math> almost everywhere, then <math>\int f=\int 0=0</math>. This motivates our definition of <math>L^1(\mu)</math> to be the set of integrable functions up to equivalence to sets of measure zero. | 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 <math>\mathbb{R}</math>, 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 <math>f=0</math> almost everywhere, then <math>\int f=\int 0=0</math>. This motivates our definition of <math>L^1(\mu)</math> to be the set of integrable functions up to equivalence to sets of measure zero. | ||
==<math>L^1(\mu)</math> Space== | |||
In this section, we will construct <math>L^1(\mu)</math>. We first show the following theorem: | In this section, we will construct <math>L^1(\mu)</math>. We first show the following theorem: | ||
==References== | ==References== | ||
Revision as of 08:32, 15 December 2020
Introduction
Let be a measure space. From our study of integration, we know that if are integrable functions, the following functions are also integrable:
- , 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 that if almost everywhere, then . 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 . We first show the following theorem: