1 Extended real numbers and their integral
TODO: describe the extended real numbers and the conventions used for their operations.
The set of extended non-negative real numbers is \(\mathbb {R}_{+,\infty } = \mathbb {R}_{+} \cup \{ +\infty \} \).
The set of extended real numbers is \(\overline{\mathbb {R}} = \mathbb {R} \cup \{ +\infty , -\infty \} \).
TODO: addition and multiplication conventions
The logarithm as a function from \(\mathbb {R}_{+, \infty }\) to \(\overline{\mathbb {R}}\) is defined as
A good property of the conventions chosen for addition and multiplication is that the usual properties of logarithm hold. In particular, for all \(x, y \in \mathbb {R}_{+, \infty }\), \(\log (xy) = \log (x) + \log (y)\).
TODO: positive and negative parts of an extended real number
For \(f : \Omega \to \overline{\mathbb {R}}\) and \(\mu \) a measure on \(\Omega \), we define the extended integral of \(f\) with respect to \(\mu \) as
where \(f^{+} = \max (f, 0)\) and \(f^{-} = -\min (f, 0)\), and their integrals are Lebesgue integrals, possibly taking the value \(+\infty \).