A weight function is a mathematical device used when performing a sum, integral, or average in order to give some elements more of a "weight" than others. They occur frequently in statistics and analysis, and are closely related to the concept of a measure. Weight functions can be constructed in both discrete and continuous settings.

Discrete weights

In the discrete setting, a weight function is a positive function defined on a discrete set A, which is typically finite or countable. The weight function w(a): = 1 corresponds to the unweighted situation in which all elements have equal weight. One can then use apply this weight to various concepts as follows:

1. If is a real-valued function, then the unweighted sum of f on A is ; but if one introduces a weight function , then one can also form the weighted sum . One common application of weighted sums arises in numerical integration.
2. If B is a finite subset of A, one can replace the unweighted cardinality |B| of B by the weighted cardinality
3. If A is a finite non-empty set, one can replace the unweighted mean or average by the weighted mean or weighted average Weighted means are commonly used in statistics to compensate for the presence of bias.

The terminology weight function arises from mechanics: if one has a collection of n objects on a lever, with weights (where weight is now interpreted in the physical sense) and locations , then the lever will be in balance if the fulcrum of the lever is at the center of mass , which is also the weighted average of the positions x_i.

Continuous weights

In the continuous setting, a weight is a positive measure such as w(x) dx on some domain Ω, which is typically a subset of an Euclidean space , for instance Ω could be an interval [a,b]. Here dx is Lebesgue measure and is a non-negative measurable function. In this context, the weight function w(x) is sometimes referred to as a density.

1. If is a real-valued function, then the unweighted integral can be generalized to the weighted integral . Note that one may need to require f to be absolutely integrable with respect to the weight w(x) dx in order for this integral to be finite.
2. If E is a subset of Ω, then the volume vol(E) of E can be generalized to the weighted volume .
3. If Ω has finite non-zero weighted volume, then we can replace the unweighted average by the weighted average
4. If and are two functions, one can generalize the unweighted inner product to a weighted inner product . See the entry on Orthogonality for more details.