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Measure and Integration Theory
Published in Athanasios Christou Micheas, Theory of Stochastic Objects, 2018
Remark 3.19 (Lebesgue and Riemann integrals) Clearly, the Lebesgue integral (when defined) can be thought of as a generalization of the Riemann integral. For what follows in the rest of the book, Lebesgue integrals will be computed using their Riemann versions, whenever possible, since Riemann integrals are easier to compute. Let f:Ω→[a,b],Ω⊂Rp, be a bounded, measurable function in the space (Ω,ℬp,μp). The following results clarify when the two integrals coincide.
Necessary conditions of extremum for functionals
Published in Simon Serovajsky, Optimization and Differentiation, 2017
We do not determine the definitions of the measurability and Lebesgue integral. Note that the measurable function is determined up to the set of zero measure. Therefore, its change on the set of zero measure (for example, on a finite or countable sets) does give a new measurable function. Hence, the elements of these functional spaces are the equivalence classes of the functions. The elements of each equivalence class are equal almost everywhere (more succinctly, a.e.) on the set Ω $ \Omega $ , that is, it can be different on a set with zero measure. We will focus in the future on the fact that some pointwise property of the functions of Lp(Ω) $ L_p(\Omega ) $ or Sobolev spaces is true a.e. on the set Ω $ \Omega $ . We shall write succinctly that this property is true on Ω $ \Omega $ .
Measurable Sets
Published in Hugo D. Junghenn, Principles of Analysis, 2018
It is illuminating to compare the construction of the two integrals in terms of how the domain [a, b] of an integrand f is partitioned. In the case of the Riemann integral, [a, b] is partitioned into subintervals [xi-1,xi] $ [x_{i-1},x_i] $ and a point xi∗ $ x_i^* $ is chosen in each. A suitable limit of the corresponding Riemann sums ∑if(xi∗)Δxi $ \sum _i f(x_i^*) \Delta x_i $ then produces the Riemann integral of f. By contrast, in the Lebesgue theory it is the range of the function that is partitioned into subintervals, these inducing, via preimages under f, a partition of [a, b]. This partition will in general not consist of intervals. However, the Lebesgue theory provides a way of “measuring” the members of the partition. The Lebesgue integral is then constructed by multiplying these measured values by (approximate) function values, summing, and taking limits.
On stability, Bohl exponent and Bohl–Perron theorem for implicit dynamic equations
Published in International Journal of Control, 2021
Do Duc Thuan, Khong Chi Nguyen, Nguyen Thu Ha, Pham Van Quoc
Let be time scale that is unbounded above. For any , the notation or means the segment on , that is or and . We can define a measure on by considering the Caratheodory construction of measures when we put . The Lebesgue integral of a measurable function f with respect to is denoted by (see Guseinov, 2003).
Pseudo-spectral optimal control of stochastic processes using Fokker Planck equation
Published in Cogent Engineering, 2019
Ali Namadchian, Mehdi Ramezani
is the initial condition, is the Weiner process, is the dimension of Weiner process, is a vector valued drift function and is a vector valued diffusion function. is the control function, which is considered to be the function of time. The first Integral in (3) is Lebesgue integral and the second one is Ito Integral. Here we consider 1-D SDE where .
Rates of return of investments whose timings are specified by a probability distribution
Published in The Engineering Economist, 2020
The integral in (6) is the Lebesgue integral calculated using the probability measure induced by X (see e.g. Durrett (2010) on the use of Lebesgue integration in probability theory). By using this type of integral one can assume that the random variables Xi are continuous or discrete, or a mixture of both.