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By Pearn W. L., Lin G. H.

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A set F is open if, for every point x ∈ F, there exists an ε-ball, B(x, ε) ⊂ F. This means that the boundary points do not belong to the set; ∂ F ⊂ F. A set G is closed if its complement G c is open. If G is closed, then ∂G ⊂ G. The sets in R are called Borel sets, and the space (R, R) is called the Borel space. The σ-algebra of Borel sets, or the Borel-σ-algebra, is defined as the σ-algebra generated by the open subsets of R; R = σ{F : F is open}. 3) An important fact is that the Borel sets can, equivalently, be generated by intervals as follows.

Thus, X is degenerate if, for some a ∈ R, P(X = a) = 1. A random variable that is not degenerate is called non-degenerate. There are different ways to interpret the equality X = Y . s. , iff X and Y are equivalent random variables, X ∼ Y . Next we provide an example to illustrate that there is a clear difference between the two equality concepts. The following example shows that two random variables, in fact, may well have the same distribution, and at the same time there is no elementary event where they agree.

The σ-algebra generated by X is defined as σ{X } = {X −1 (A) : A ∈ R} = {ω : X (ω) ∈ A} : A ∈ R . 3 Check that, equivalently, σ{X } = σ{X −1 ((−∞, x]) : x ∈ R}. 5 For a set {X k }, finite or infinite, of random variables, σ {X k }k = σ σ{X k } . 2, the complete description of the distribution of a random variable X would require knowledge about P(X ∈ A) for all sets A ∈ R. And, once again, the fact that the intervals (−∞, x] generate R comes to our rescue. This fact is manifested by introducing the concept of distribution functions.

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