The question which arises is whether, in order to define the set function Px(.) we need to consider all the elements of the borel field B. The answer is that we do not need to do that because, as argued above any such element of B can be expressed in terms of the semi-closed intervals(oo, This implies that by choosing such semi-closed intervals intelligently, we can define Px() with the minimum of effort. For example, we may define 0 X<0. P2(-∞,x]) X≤0( That is x=0) X≤1( That is r=1) X≤2(That we can see. the semi-closed intervals were chosen to divided the real line at the points corresponding to the value taken by X. This way of defining the semi-closed intervals is clearly non-unique but will prove very convenient in the next subsection In fact, the event and probability structure of(S, F, P()) is preserved in the induced probability space(R, B, Pr(). We traded S, a set of arbitrary elements for R, the real line; F a o-field of subset of S with B, the Borel field on the real line; and P() a set function defined on arbitrary sets with Px(), a set function on semi-closed intervals of the real lineThe question which arises is whether, in order to define the set function PX(·), we need to consider all the elements of the Borel field B. The answer is that we do not need to do that because, as argued above, any such element of B can be expressed in terms of the semi-closed intervals (−∞, x]. This implies that by choosing such semi-closed intervals ’intelligently’, we can define PX(·) with the minimum of effort. For example, we may define: Px((−∞, x]) =    0 X < 0, 1 4 X ≤ 0 (That is x = 0), 3 4 X ≤ 1 (That is x = 1), 1 X ≤ 2 (That is x = 2), As we can see, the semi-closed intervals were chosen to divided the real line at the points corresponding to the value taken by X. This way of defining the semi-closed intervals is clearly non-unique but will prove very convenient in the next subsection. In fact, the event and probability structure of (S, F,P(·)) is preserved in the induced probability space (R, B, Px(·)). We traded S, a set of arbitrary elements, for R, the real line; F a σ-field of subset of S with B, the Borel field on the real line; and P(·) a set function defined on arbitrary sets with PX(·), a set function on semi-closed intervals of the real line. 16