almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere"...

almost everywhere convergence of a sequence of measurable functions defined on a measurable space. That means pointwise convergence almost everywhere...

elementary real analysis. To say that the sequence Xn converges almost surely or almost everywhere or with probability 1 or strongly towards X means that P (...

set might not be empty. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample...

a function in L1 whose Fourier series diverges almost everywhere (later improved to diverge everywhere). Jean-Pierre Kahane and Yitzhak Katznelson proved...

generally, "almost all" is sometimes used in the sense of "almost everywhere" in measure theory, or in the closely related sense of "almost surely" in...

nth power almost everywhere locally is an nth power globally can fail in two distinct ways: the element can be an nth power almost everywhere locally but...

absolute value is Lebesgue integrable, where functions which agree almost everywhere are identified. More generally, let ( S , Σ , μ ) {\displaystyle (S...

functions is bounded in absolute value by an integrable function and is almost everywhere pointwise convergent to a function then the sequence converges in...

statements that are not valid for all elements in a set, but rather almost everywhere, that is, except on a set of measure zero. While the exact definition...

{\displaystyle X_{1}} and Y 2 {\displaystyle Y_{2}} ; the distribution has support everywhere but at the origin. The right panel shows the joint distribution of Y 1...

absolutely continuous and thus differentiable almost everywhere, and satisfies |f′(x)| ≤ K for almost all x in I, then f is Lipschitz continuous with...

μ {\displaystyle \mu } -almost everywhere e s s s u p p ⁡ ( f ) := X ∖ ⋃ { Ω ⊆ X : Ω  is open and  f = 0 μ -almost everywhere in  Ω } . {\displaystyle...

continuous ⊆ absolutely continuous ⊆ bounded variation ⊆ differentiable almost everywhere. A continuous function fails to be absolutely continuous if it fails...

property, then f = g {\displaystyle f=g} μ {\displaystyle \mu } -almost everywhere. The function f {\displaystyle f} is commonly written d ν d μ {\frac...

numbers α, β ≥ 0, not both of them zero, such that α|f |p = β |g|q μ-almost everywhere. The numbers p and q above are said to be Hölder conjugates of each...

conditions, the time average of a function along the trajectories exists almost everywhere and is related to the space average. Two of the most important theorems...

points. Weierstrass's demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning several proofs...

Exiled from Almost Everywhere (Spanish: El exiliado de aquí y allá) is a 2008 novel by the Spanish writer Juan Goytisolo. It tells the story of a Parisian...

despite being differentiable almost everywhere. This is because any weak derivative of c would have to be equal almost everywhere to the classical derivative...

derivative, and measure. Although it is continuous everywhere, and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument...

respects the equivalence relation of almost-everywhere equality. If f and g are functions such that f = g almost everywhere, then f is Lebesgue integrable if...

pointwise almost everywhere, by the Titchmarsh theorem. In the case p = 1, the Hilbert transform still converges pointwise almost everywhere, but may itself...

derivative almost everywhere. The Minkowski question-mark function is continuous and strictly increasing but has zero derivative almost everywhere. Satisfaction...

nonconstant periodic solutions lying entirely within the region. "Almost everywhere" means everywhere except possibly in a set of measure 0, such as a point or...

function, but to equivalence classes of functions that are equal almost everywhere. The square integrable functions (in the sense mentioned in which...

result in mathematical analysis establishing the (Lebesgue) pointwise almost everywhere convergence of Fourier series of L2 functions, proved by Lennart Carleson (1966)...

can conclude that the function F is differentiable almost everywhere and F′(x) = f(x) almost everywhere. On the real line this statement is equivalent to...

lemma remains true if its assumptions hold μ {\displaystyle \mu } -almost everywhere. In other words, it is enough that there is a null set N {\displaystyle...

the most numerous Italian armoured fighting vehicle and saw service almost everywhere the Italians fought in the Second World War but proved inadequate...