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Lattice Theory
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
One connection of measure theory and other mathematical systems is via functions. For a particular example, suppose that X is a measurable space, (Y,τ) is a topological space, and f:X→Y is a function, then f is called a measurable function if and only if f−1(U) is a measurable set in X∀U∈τ. The proof of the following theorem is an easy consequence of the theorem described in Exercise 2.2.3(2).
Stochastic differential equations
Published in Alfio Borzì, Modelling with Ordinary Differential Equations, 2020
Now, we can define a random variable as follows. A real-valued random variable y:Ω→R is a measurable function from the sample space Ω of the probability space (Ω,F,P) to the space R equipped with its Borel σ-algebra.If y can take uncountably infinite many values, then it is called a continuous (space) random variable. If the image of y is finite or countably infinite, then we have a discrete (space) random variable.
Basic Stochastic Mathematics
Published in Ning Zhang, Chongqing Kang, Ershun Du, Yi Wang, Analytics and Optimization for Renewable Energy Integration, 2019
Ning Zhang, Chongqing Kang, Ershun Du, Yi Wang
Random variables represent real single-value functions of various results of random tests. The possible values of the random variables are outcomes of a random phenomenon. A random variable X can be viewed as a measurable function from a set of possible outcomes Ω to a measurable space E. Mathematically, the probability that X takes on a value in a measurable set S⊆E is written as: () Pro(X∈S)=Pro({ω∈Ω|X(ω)∈S})
On deferred-statistical convergence of uncertain fuzzy sequences
Published in International Journal of General Systems, 2022
L. Nayak, B. C. Tripathy, P. Baliarsingh
Here triplet is called an uncertainty space whereas, the element Λ in is an event of it. A measurable function ϱ from an uncertainty space to is called an uncertain variable, that is, for any Borel set B of real numbers, the set is an event. For an uncertain variable ϱ, its uncertainty distribution ψ is given by The expected value of an uncertain variable ϱ is defined by if at least one of the above two integrals exists and finite.
Quasidifferentiabilities of the expectation functions of random quasidifferentiable functions
Published in Optimization, 2020
Sida Lin, Ming Huang, Zunquan Xia, Dan Li
Given the probability space , a set is called measurable if . A mapping is called measurable if for any Borel set , its inverse image is measurable. A measurable mapping is called a random vector. A measurable function is called a random variable.
Optimal control for uncertain random continuous-time systems
Published in Optimization, 2023
Let be an uncertainty space, and a probability space. The product is called a chance space established by [41]. An uncertain random variable is a measurable function from a chance space to the set of real numbers. The chance measure, chance distribution, and expected value of an uncertain random variable are defined in [41].