Mass measured in grams is a function from this collection of weight to positive real numbers. The term "weight function", an allusion to this example, is used in pure and applied mathematics.
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.
Let be the set of all functions from a setX to real numbers . Because is a field, may be turned into a vector space and a commutative algebra over the reals with the following operations:
These operations extend to partial functions from X to with the restriction that the partial functions f + g and fg are defined only if the domains of f and g have a nonempty intersection; in this case, their domain is the intersection of the domains of f and g.
Also, since is an ordered set, there is a partial order
The σ-algebra of Borel sets is an important structure on real numbers. If X has its σ-algebra and a function f is such that the preimagef−1(B) of any Borel set B belongs to that σ-algebra, then f is said to be measurable. Measurable functions also form a vector space and an algebra as explained above in § Algebraic structure.
Moreover, a set (family) of real-valued functions on X can actually define a σ-algebra on X generated by all preimages of all Borel sets (or of intervals only, it is not important). This is the way how σ-algebras arise in (Kolmogorov's) probability theory, where real-valued functions on the sample spaceΩ are real-valued random variables.
Continuous functions also form a vector space and an algebra as explained above in § Algebraic structure, and are a subclass of measurable functions because any topological space has the σ-algebra generated by open (or closed) sets.
A measure on a set is a non-negative real-valued functional on a σ-algebra of subsets.[2]Lp spaces on sets with a measure are defined from aforementioned real-valued measurable functions, although they are actually quotient spaces. More precisely, whereas a function satisfying an appropriate summability condition defines an element of Lp space, in the opposite direction for any f ∈ Lp(X) and x ∈ X which is not an atom, the value f(x) is undefined. Though, real-valued Lp spaces still have some of the structure described above in § Algebraic structure. Each of Lp spaces is a vector space and have a partial order, and there exists a pointwise multiplication of "functions" which changes p, namely
For example, pointwise product of two L2 functions belongs to L1.