ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Binary cartesian set product
Binary relation
Map
Simple map
Simple function
Measurable simple complex function
Simple integral
Unsigned basic integral
P-integrable basic function
Absolutely integrable function
Definition D1748
Signed basic integral
Formulation 0
Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that
(i) $f : X \to [-\infty, \infty]$ is an D1921: Absolutely integrable function on $M$
The signed integral of $f$ with respect to $M$ is the D993: Real number \begin{equation} \int_X f \, d \mu := \int_X f^+ \, d \mu - \int_X f^- \, d \mu \end{equation}
Formulation 1
Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that
(i) $f : X \to [-\infty, \infty]$ is an D1921: Absolutely integrable function on $M$
The signed integral of $f$ with respect to $M$ is the D993: Real number \begin{equation} \mu(f) : = \mu(f^+) - \mu(f^-) \end{equation}
Children
Complex integral
Results
R4500
R4513
R4512
Basic integral of almost everywhere zero function is zero
Basic integral over a set of measure zero is zero
Isotonicity of signed basic integral
Signed basic expectation with respect to a point probability measure
Signed basic integral with respect to a point measure