Bounded signed integrals obey the density formula

Given: A positive measure $\mu$, a finite nonnegative measurable density $p$, the probability measure $P=pd\mu$, and a measurable real function $f$ with $\mathrm{\mid }f\mathrm{\mid }\le M<\mathrm{\infty }$.

The measure with density $f$ relative to $\mu$: $P(A)={\int }_{A}pd\mu$ for every measurable $A$.

Integrating against a density agrees with integrating the product: for nonnegative measurable $g$, $\int gdP=\int gpd\mu$.

Closure properties of measurable functions used by the integral: the positive and negative parts $f^{+},f^{-}$ of a real measurable $f$ are measurable.

Arithmetic and lattice operations preserve measurability whenever they are defined: the product of finite real measurable functions is measurable.

Monotonicity and nonnegative homogeneity of the nonnegative integral: the nonnegative integral is monotone and homogeneous for nonnegative scalars, with the integral of the zero function equal to zero.

Integrable real and complex functions, and their integrals: $f$ is integrable when $\int \mathrm{\mid }f\mathrm{\mid }$ is finite, and its integral is $\int f^{+}-\int f^{-}$.

The Lebesgue integral is linear on $L^1(\mu )$: integrable functions have finite linear integrals.

Probability measures and probability spaces: $P(X)=1$.