vast class of sets, the so-called measurable sets. Indeed, non-measurable sets almost never occur in applications. Measurable sets, given in a measurable space by definition, lead to measurable functions and maps. In order to turn a topological space into a measurable space one endows it with a The of Borel sets is the most popular, but not the only choice. (Baire sets, universally measurable sets, etc, are also used sometimes.) The topology is not uniquely determined by the Borel for example, the norm topology and the weak topology on a separable Hilbert space lead to the same Borel . Not every is the Borel of some topology. Actually, a can be generated by a given collection of sets (or functions) irrespective of any topology. Every subset of a measurable space is itself a measurable space. Standard measurable spaces (also called standard Borel spaces) are especially useful due to some similarity to compact spaces (see EoM). Every bijective measurable mapping between standard measurable spaces is an isomorphism; that is, the inverse mapping is also measurable. And a mapping between such spaces is measurable if and only if its graph is measurable in the product space. Similarly, every bijective continuous mapping between compact metric spaces is a homeomorphism; that is, the inverse mapping is also continuous. And a mapping between such spaces is continuous if and only if its graph is closed in the product space. Every Borel set in a Euclidean space (and more generally, in a complete separable metric space), endowed with the Borel is a standard measurable space. All uncountable standard measurable spaces are mutually isomorphic. A measure space is a measurable space endowed with a measure. A Euclidean space with the Lebesgue measure is a measure space. Integration theory defines integrability and integrals of measurable functions on a measure space. Sets of measure 0, called null sets, are negligible. Accordingly, a "mod 0 isomorphism" is defined as isomorphism between subsets of full measure (that is, with negligible complement). A probability space is a measure space such that the measure of the whole space is equal to 1. The product of any family (finite or not) of probability spaces is a probability space. In contrast, for measure spaces in general, only the product of finitely many spaces is defined. Accordingly, there are many infinite-dimensional probability measures (especially, Gaussian measures), but no infinite-dimensional Lebesgue measures. Standard probability spaces are especially