(i) | $f : X \to [-\infty, \infty]$ is an D1921: Absolutely integrable function on $M$ |

**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}

Definition D1748

Signed basic integral

Formulation 0

Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that

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}

(i) | $f : X \to [-\infty, \infty]$ is an D1921: Absolutely integrable function on $M$ |

Formulation 1

Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that

The **signed integral** of $f$ with respect to $M$ is the D993: Real number
\begin{equation}
\mu(f) : = \mu(f^+) - \mu(f^-)
\end{equation}

(i) | $f : X \to [-\infty, \infty]$ is an D1921: Absolutely integrable function on $M$ |

Children

▶ | D1749: Complex integral |

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