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Definition D1748
Signed basic integral

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 $$\int_X f \, d \mu := \int_X f^+ \, d \mu - \int_X f^- \, d \mu$$

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 $$\mu(f) : = \mu(f^+) - \mu(f^-)$$
Children
 ▶ D1749: Complex integral
Results
 ▶ R4500 ▶ R4513 ▶ R4512 ▶ R1903: Basic integral of almost everywhere zero function is zero ▶ R4562: Basic integral over a set of measure zero is zero ▶ R1514: Isotonicity of signed basic integral ▶ R4662: Signed basic expectation with respect to a point probability measure ▶ R3863: Signed basic integral with respect to a point measure