Measure space and integral

A measure space is a triple $(X,\mathrm{\Sigma },\mu )$ where $\mathrm{\Sigma }$ contains $\mathrm{\varnothing },X$ and is closed under complements and countable unions (a $\sigma$-algebra) of subsets of $X$ and $\mu :\mathrm{\Sigma }\to [0,\mathrm{\infty }]$ is countably additive with $\mu (\mathrm{\varnothing })=0$. A measurable function $f:X\to [-\mathrm{\infty },\mathrm{\infty }]$ has measurable inverse images of Borel sets, where $[-\mathrm{\infty },\mathrm{\infty }]$ has its order topology: finite open intervals and the endpoint neighborhoods $[-\mathrm{\infty },a)$ and $(a,\mathrm{\infty }]$ form a basis. Its Borel sigma-algebra is generated by these open sets. A complex-valued function is measurable exactly when its finite real and imaginary parts are measurable, equivalently for the usual Borel structure of $\mathbb{C}$. Almost-everywhere assertions mean outside a measurable $\mu$-null set; completeness of $\mu$ is not assumed. Nonnegative extended addition uses $a+\mathrm{\infty }=\mathrm{\infty }+a=\mathrm{\infty }$ for $a\in [0,\mathrm{\infty }]$; positive scalars multiply $\mathrm{\infty }$ to $\mathrm{\infty }$, while $0\cdot \mathrm{\infty }=0$. Signed expressions $\mathrm{\infty }-\mathrm{\infty }$ remain undefined and are not used in signed integrals. A nonnegative simple function has the form $s={\sum }_{j=1}^m{c}_j{\mathbf{1}}_{A_j}$ with finitely many disjoint $A_j\in \mathrm{\Sigma }$ and finite $c_j\ge 0$. Define $c\mu (A)=+\mathrm{\infty }$ if $c>0$ and $\mu (A)=+\mathrm{\infty }$, define $0\cdot \mathrm{\infty }:=0$, and define the finite sum of these extended nonnegative terms to be $+\mathrm{\infty }$ if any term is $+\mathrm{\infty }$ and otherwise the real finite sum of Finite sums and finite products, by recursion. Then $\int sd\mu :={\sum }_j{c}_j\mu (A_j)$; using canonical level sets $\{s=c\}$ makes this value representation-independent. For measurable $f\ge 0$, define $\int fd\mu :=\sup \{\int sd\mu :s\text{ is simple and }0\le s\le f\}$, using Complete ordered field (least-upper-bound property) for a bounded set of finite values and setting the supremum to $+\mathrm{\infty }$ when the values are unbounded or one is infinite. A real measurable $f$ is integrable when $\int \mathrm{\mid }f\mathrm{\mid }d\mu <\mathrm{\infty }$, and then $\int f:=\int f^{+}-\int f^{-}$, where $f^{+}=\max (f,0)$ and $f^{-}:=\max (-f,0)$. Complex integrability and integration are defined by integrable real and imaginary parts.

The integral may equal $+\mathrm{\infty }$ for a nonnegative function. For an integrable signed or complex function it is finite. The measure $\mu$ is the reference measure: changing it can change an integral and any entropy defined from a density relative to it.