if is Lebesgue measure on and is the property of not being equal to (i.e. is true if and only if ), then each holds almost everywhere, but the conjunction does not hold anywhere. As a consequence of the first two properties, it is often possible to reason about "almost every point" of a measure space as though it were an ordinary point rather than an abstraction. This is often done implicitly in informal mathematical arguments. However, one must be careful with this mode of reasoning because of the third bullet above: universal quantification over uncountable families of statements is valid for ordinary points but not for "almost every point". Examples If f : R → R is a Lebesgue integrable function and almost everywhere, then for all real numbers with equality if and only if almost everywhere. If f : [a, b] → R is a monotonic function, then f is differentiable almost everywhere. If f : R → R is Lebesgue measurable and for all real numbers , then there exists a set E (depending on f) such that, if x is in E, the Lebesgue mean converges to as decreases to zero. The set E is called the Lebesgue set of f. Its complement can be proved to have measure zero. In other words, the Lebesgue mean of f converges to f almost everywhere. A bounded function is Riemann integrable if and only if it is continuous almost everywhere. As a curiosity, the decimal expansion of almost every real number in the interval [0, 1] contains the complete text of Shakespeare's plays, encoded in ASCII; similarly for every other finite digit sequence, see Normal number. Definition using ultrafilters Outside of the context of real analysis, the notion of a property true almost everywhere is sometimes defined in terms of an ultrafilter. An ultrafilter on a set X is a maximal collection F of subsets of X such that: If U ∈ F and U ⊆ V then V ∈ F The intersection of any two sets in F is in F The empty set is not in F A property P of points in X holds almost everywhere, relative to an ultrafilter F, if the set of points for which P holds is in F. For example, one construction of the hyperreal number system defines a hyperreal number as an equivalence class of sequences that are equal