Extendedvalued

Extendedvalued is a term used in mathematics to describe objects whose value range is allowed to include an infinite value. It commonly appears in three related contexts: extended real-valued functions, extended-valued measures, and extended valuations in field theory. In each case, the additional infinity value serves to compactly describe limiting behavior, divergence, or zero elements.

In analysis, an extended-valued function is a map f from a set X to the extended real line, typically denoted R ∪ {±∞}. This allows f to take the values +∞ or −∞. Such functions are convenient for describing limits and improper integrals. When f is nonnegative and measurable, the integral ∫ f dμ is defined as an extended real number, which may be finite or infinite.

In measure theory, measures can be extended to take infinite values. A measure μ on a measurable space may satisfy μ(X) = ∞ for the whole space X. Extended-valued measures enable the formulation of integral and probability statements in cases where total mass is unbounded. The extended real line is also used to express convergence, tails, and integrals that diverge to infinity.

In valuation theory, an extended-valued valuation v: K → Γ ∪ {∞} assigns ∞ to zero and obeys v(xy) = v(x) + v(y) and v(x + y) ≥ min{v(x), v(y)} when applicable. Here Γ is a totally ordered value group, and ∞ acts as an absorbing top element. Extended valuations are widely used in algebraic and number-theoretic contexts.

See also extended real numbers, valuation, and measure theory.