using calligraphic capital letters, or the Fraktur typeface. Thus may be denoted as or Particular cases and examples Separable σ-algebras A separable -algebra (or separable -field) is a -algebra that is a separable space when considered as a metric space with metric for and a given measure (and with being the symmetric difference operator). Note that any -algebra generated by a countable collection of sets is separable, but the converse need not hold. For example, the Lebesgue -algebra is separable (since every Lebesgue measurable set is equivalent to some Borel set) but not countably generated (since its cardinality is higher than continuum). A separable measure space has a natural pseudometric that renders it separable as a pseudometric space. The distance between two sets is defined as the measure of the symmetric difference of the two sets. Note that the symmetric difference of two distinct sets can have measure zero; hence the pseudometric as defined above need not to be a true metric. However, if sets whose symmetric difference has measure zero are identified into a single equivalence class, the resulting quotient set can be properly metrized by the induced metric. If the measure space is separable, it can be shown that the corresponding metric space is, too. Simple set-based examples Let be any set. The family consisting only of the empty set and the set called the minimal or trivial σ-algebra over The power set of called the discrete σ-algebra. The collection is a simple σ-algebra generated by the subset The collection of subsets of which are countable or whose complements are countable is a σ-algebra (which is distinct from the power set of if and only if is uncountable). This is the σ-algebra generated by the singletons of Note: "countable" includes finite or empty. The collection of all unions of sets in a countable partition of is a σ-algebra. Stopping time sigma-algebras A stopping time can define a -algebra the so-called stopping time sigma-algebra, which in a filtered probability space describes the information up to the random time in the sense that, if the filtered probability space is interpreted as a random experiment, the maximum information that can be found out about the experiment from arbitrarily often repeating it until the time is σ-algebras generated by families of sets σ-algebra generated by an arbitrary family Let be an arbitrary family of subsets of Then there exists a unique smallest