In mathematics, a Loeb space is a type of measure space introduced by Loeb (1975) using non-standard analysis.
Loeb's construction starts with a finitely additive map ν from an internal algebra A of sets to the non-standard reals. Define μ to be given by the standard part of ν, so that μ is a finitely additive map from A to the extended reals R∪∞∪–∞. Define M to be the σ-algebra generated by A. Then μ extends to a countably additive measure on M, called a Loeb measure.
Even if ω is a non-standard measure and A is a non-standard σ-algebra, the algebra A need not be an ordinary σ-algebra as it need not be closed under countable unions, so it is still necessary to take M to be the σ-algebra generated by A rather than A itself.