# Basic Ideas

• Sergey V. Lototsky
• Boris L. Rozovsky
Chapter
Part of the Universitext book series (UTX)

## Abstract

Given a probability space $$(\varOmega,\mathcal{F}, \mathbb{P})$$ and two measurable spaces, $$(A,\mathcal{A})$$ and $$(B,\mathcal{B})$$, a random function X is a (measurable) mapping from $$(A\times \varOmega,\mathcal{A}\times \mathcal{F})$$ to $$(B,\mathcal{Y})$$. In the traditional terminology, the random process corresponds to $$A,B \subset \mathbb{R}$$; a random field corresponds to $$A = \mathbb{R}^{\mathrm{d}}$$, $$B = \mathbb{R}$$. A sample path , or sample trajectory of X is the function X(⋅ , ω) for fixed ωΩ. A modification of X is a random function $$\bar{X}$$ such that, for every aA, $$\mathbb{P}\big(X(a) =\bar{ X}(a)\big) = 1$$; note that this, in general, DOES NOT mean that $$\mathbb{P}\big(X(a) =\bar{ X}(a)\ \text{for all}\ a \in A\big) = 1$$ (although it does if both X and $$\bar{X}$$ are continuous).

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## Authors and Affiliations

• Sergey V. Lototsky
• 1
• Boris L. Rozovsky
• 2
1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA
2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA