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Measure and Integration Theory
Published in Athanasios Christou Micheas, Theory of Stochastic Objects, 2018
Almost everywhere A property is said to hold almost everywhere with respect to μ and we write a.e. [μ], if the set of points where it fails to hold is a set of measure zero, i.e., if the property is expressed in terms of a collection of ω ∈ Ω forming the set A, then A a.e. [μ] if and only if μ(Αc) = 0. If the measure μ is understood by the context, we simply say that the property holds a.e. If μ(Αc) = 0 for some A∈A, then A is a support of μ and μ is concentrated on A. For a finite measure μ, A is a support if and only if μ(Α) = μ(Ω).
Probability Theory
Published in Hugo D. Junghenn, Principles of Analysis, 2018
Many terms in modern probability theory reflect the classical origins of the subject as well as its use in analysing real data. For example, the set Ω $ \Omega $ of a probability space (Ω,F,P) $ (\Omega ,\boldsymbol{{ \fancyscript {F}}}, P) $ is called the sample space (in practice, the set of outcomes of an experiment), and members of F $ \boldsymbol{{ \fancyscript {F}}} $ are called events (sets of outcomes). Properties holding almost everywhere are said to hold almost surely (a.s.). A real-valued (Borel) measurable function is called a random variable and may be viewed as a numerical description of an outcome of an experiment. A measurable function that takes values in Rd $ \mathbb R ^{d} $ is called a d-dimensional random variable.
Wavelet Transforms
Published in Wai-Kai Chen, Mathematics for Circuits and Filters, 2000
P.P. Vaidyanathan, Igor Djokovic
When something is said to be true “almost everywhere” (abbreviated a.e.) or “for almost all t” it means that the statement holds everywhere, except possibly on a set of measure zero. For example, if x(t) = y(t) everywhere except for integer values of t, then x(t) = y(t) a.e. An important fact in Lebesgue integration theory is that if two Lebesgue integrable functions are equal a.e., then their integrals are equal. In particular, if x(t) = 0 a.e., the Lebesgue integral ∫ x(t)dt exists and is equal to zero.
Proper efficiency, scalarization and transformation in multi-objective optimization: unified approaches
Published in Optimization, 2022
Moslem Zamani, Majid Soleimani-damaneh
According to Rademacher's theorem, every locally Lipschitz function on is almost everywhere differentiable in the sense of Lebesgue measure [9]. Let be a locally Lipschitz function. The generalized Jacobian of g at , denoted by , is defined by where is the set of points at which g is not differentiable, and is the Jacobian matrix of g at . If g is continuously differentiable at , then . See [9] for more information about the generalized Jacobian. Hereafter, for a measurable set , denotes its Lebesgue measure.
Generalized desirability functions: a structural and topological analysis of desirability functions
Published in Optimization, 2020
Başak Akteke-Öztürk, Gerhard-Wilhelm Weber, Gülser Köksal
These max-type functions are nondifferentiable but Lipschitz continuous by Hager's Theorem [18] and they are almost everywhere differentiable in the sense of Lebesgue measure, i.e. a set of Lebesgue-measure 0 by Rademacher's Theorem [19]. Moreover, they are positive and bounded by 0 from below, piecewise smooth generically and not necessarily convex. Although they may lack convexity, we can call the structure of them as convex-like since, via a pathfollowing [20] along y, we select as active the function which has the biggest slope (derivative).
Guaranteed a posteriori estimation of unknown right-hand sides of linear periodic systems of ODEs
Published in Applicable Analysis, 2022
Oleksandr Nakonechnyi, Yuri Podlipenko
By a solution of this problem, we mean a function that satisfies Equation (1) almost everywhere (a.e.) on (except on a set of Lebesgue measure 0) and the conditions (2). Here is the space of functions absolutely continuous on an interval for which the derivative that exists almost everywhere on belongs to space