Editing Weight function (section)

== Continuous weights ==
In the continuous setting, a weight is a positive [[measure (mathematics)|measure]] such as <math>w(x) \, dx</math> on some [[domain (mathematical analysis)|domain]] <math>\Omega</math>, which is typically a [[subset]] of a [[Euclidean space]] <math>\R^n</math>, for instance <math>\Omega</math> could be an [[Interval (mathematics)|interval]] <math>[a,b]</math>.  Here <math>dx</math> is [[Lebesgue measure]] and <math>w\colon \Omega \to \R^+</math> is a non-negative [[measurable]] [[mathematical function|function]].  In this context, the weight function <math>w(x)</math> is sometimes referred to as a [[density]].

=== General definition ===
If <math>f\colon \Omega \to \R</math> is a [[real number|real]]-valued [[mathematical function|function]], then the ''unweighted'' [[integral]]

:<math>\int_\Omega f(x)\ dx</math>

can be generalized to the ''weighted integral'' 

:<math>\int_\Omega f(x) w(x)\, dx</math>

Note that one may need to require <math>f</math> to be [[absolutely integrable function|absolutely integrable]] with respect to the weight <math>w(x) \, dx</math> in order for this integral to be finite.

=== Weighted volume ===
If ''E'' is a subset of <math>\Omega</math>, then the [[volume]] vol(''E'') of ''E'' can be generalized to the ''weighted volume'' 
:<math> \int_E w(x)\ dx,</math>

=== Weighted average ===
If <math>\Omega</math> has finite non-zero weighted volume, then we can replace the unweighted [[average]] 

:<math>\frac{1}{\mathrm{vol}(\Omega)} \int_\Omega f(x)\ dx</math>

by the '''weighted average''' 

:<math> \frac{\displaystyle\int_\Omega f(x)\, w(x) \, dx}{\displaystyle\int_\Omega w(x) \, dx}</math>

=== Bilinear form ===
If <math> f\colon \Omega \to {\mathbb R}</math> and <math> g\colon \Omega \to {\mathbb R}</math> are two functions, one can generalize the unweighted [[bilinear form]] 

:<math>\langle f, g \rangle := \int_\Omega f(x) g(x)\ dx</math>

to a weighted bilinear form 

:<math>{\langle f, g \rangle}_w := \int_\Omega f(x) g(x)\ w(x)\ dx.</math>

See the entry on [[orthogonal polynomials]] for examples of weighted [[orthogonal functions]].