Norm of an Integrable Function For any integrable function
, we define the norm of as In particular, the norm is the integral of the absolute value of .
Space We denote by
the normed vector space of all integrable functions on , modulo the equivalence relation of being equal almost everywhere, with the norm defined by the norm.
Proposition
The
norm is a norm on :
- For all
, . . if and only if a.e.
Riesz-Fischer Theorem
The space
is complete with respect to the norm.
Corollary
If
converges to in , then there exists a subsequence such that
Proof Any convergent sequence (in a metric space) is Cauchy, and hence has a Cauchy subsequence. By the completeness of
Dense
We say that a family
of integrable functions is dense in if for every and every , there exists such that .
e.g. The following families are dense in
- The simple functions.
- The step functions.
- The continuous functions with compact support.
Invariance Properties
Convolution
Convolution
The convolution of two functions
is defined by
Proposition
Convolution is commutative, associative, and distributive over addition.
Fubini’s Theorem
Fubini's Theorem
Suppose
is integrable on . Then for almost every , the slice is integrable on , the function defined by is integrable on . Moreover:
Tonelli's Theorem
Suppose
is non-negative and measurable on . Then for almost every , the slice is measurable on , and the function defined by is measurable on . Moreover: in the extended sense.