σ-algebra which contains every set in (even though may or may not itself be a σ-algebra). It is, in fact, the intersection of all σ-algebras containing (See intersections of σ-algebras above.) This σ-algebra is denoted and is called the σ-algebra generated by If is empty, then Otherwise consists of all the subsets of that can be made from elements of by a countable number of complement, union and intersection operations. For a simple example, consider the set Then the σ-algebra generated by the single subset is By an abuse of notation, when a collection of subsets contains only one element, may be written instead of in the prior example instead of Indeed, using to mean is also quite common. There are many families of subsets that generate useful σ-algebras. Some of these are presented here. σ-algebra generated by a function If is a function from a set to a set and is a -algebra of subsets of then the -algebra generated by the function denoted by is the collection of all inverse images of the sets in That is, A function from a set to a set is measurable with respect to a σ-algebra of subsets of if and only if is a subset of One common situation, and understood by default if is not specified explicitly, is when is a metric or topological space and is the collection of Borel sets on If is a function from to then is generated by the family of subsets which are inverse images of intervals/rectangles in A useful property is the following. Assume is a measurable map from to and is a measurable map from to If there exists a measurable map from to such that for all then If is finite or countably infinite or, more generally, is a standard Borel space (for example, a separable complete metric space with its associated Borel sets), then the converse is also true. Examples of standard Borel spaces include with its Borel sets and with the cylinder σ-algebra described below. Borel and Lebesgue σ-algebras An important example is the Borel algebra over any topological space: the σ-algebra generated by the open sets (or, equivalently, by the closed sets). Note that this σ-algebra is not, in general, the whole power set. For a non-trivial example that is not a Borel set, see the Vitali set or Non-Borel sets. On the Euclidean space another σ-algebra is