Introduction to the Mathematical and Statistical Foundations of Econometrics
Outer Measure
Any subset A of [0,1] can always be completely covered by a finite or countably infinite union of sets in the algebra ^0: A c U°=j Aj, where Aj є ^0; hence, the “probability” of A is bounded from above by P(Aj). Taking the infimum of 2^= P (Aj) over all countable sequences of sets Aj є such that A c Ucj=1 Aj then yields the outer measure:
Definition 1.7: Let Ж0 be an algebra of subsets of The outer measure of an
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arbitrary subset A of ^ is
Note that it is not required in (1.21) that Uj=1 Aj є ^c.
Because a union of sets Aj in an algebra can always be written as a union of disjoint sets in the algebra (see Theorem 1.4), we may without loss of generality assume that the infimum in (1.21) is taken over all disjoint sets Aj in such that A c Uj>=1 Aj. This implies that
If A є then P*(A) = P(A). (1.22)
The question now arises, For which other subsets of ^ is the outer measure a probability measure? Note that the conditions (1.8) and (1.9) are satisfied for the outer measure P* (Exercise: Why?), but, in general, condition (1.10) does not
hold for arbitrary sets. See, for example, Royden (1968,63-64). Nevertheless, it is possible to extend the outer measure to a probability measure on a a-algebra &containing &0:
Theorem 1.9: Let P be a probability measure on {^, &0}, where &0 is an algebra, and let & = a (&0) be the smallest a-algebra containing the algebra &0. Then the outer measure P* is a unique probability measure on {^, &}, which coincides with P on &0.
The proof that the outer measure P* is a probability measure on & = a (&0) that coincides with P on &0 is lengthy and is therefore given in Appendix I. B. The proof of the uniqueness of P* is even longer and is therefore omitted.
Consequently, for the statistical experiment under review there exists a a - algebra & of subsets of ^ = [0, 1] containing the algebra &0 defined in (1.20) for which the outer measure P*: & ^ [0, 1] is a unique probability measure. This probability measure assigns its length as probability in this case to each interval in [0, 1]. It is called the uniform probability measure.
It is not hard to verify that the a-algebra & involved contains all the Borel subsets of [0, 1]: {[0, 1] П B, for all Borel sets B} c &. (Exercise: Why?) This collection of Borel subsets of [0, 1] is usually denoted by [0, 1] П B and is a a-algebra itself (Exercise: Why?). Therefore, we could also describe the probability space of this statistical experiment by the probability space {[0, 1], [0, 1] П B, P*}, where P* is the same as before. Moreover, defining the probability measure г on B as /x(B) = P*([0, 1] П B), we can also describe this statistical experiment by the probability space {R, B, г}, where, in particular
г((—ж, x ]) = 0 г((-то, x ]) = x г((-то, x ]) = 1
and, more generally, for intervals with endpoints a < b,
ti((a, b)) = ti([a, b]) = ti([a, b)) = fi((a, b]) = г((-то, b]) — г((-то, a]),
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whereas for all other Borel sets B,
Lebesgue Measure and Lebesgue Integral
1.7.1. Lebesgue Measure
Along similar lines as in the construction of the uniform probability measure we can define the Lebesgue measure as follows. Consider a function X that assigns its length to each open interval (a, b), X((a, b)) = b — a, and define for all other Borel sets B in R,
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X(B) = inf V X((aj, bj)) = inf V(bj — aj).
BcU“1(aJ.,bj) 7 7 BcU“=1(aJ. bj) j~{ j 7
for all other Borel sets B in Rk. This is the Lebesgue measure on Rk, which measures the area (in the case k = 2) or the volume (in the case k > 3) of a Borel set in Rk, where again the measurement is taken from the outside.
Note that, in general, Lebesgue measures are not probability measures because the Lebesgue measure can be infinite. In particular, X(Rk) = oo. However, if confined to a set with Lebesgue measure 1, this measure becomes the uniform probability measure. More generally, for any Borel set A є Rk with positive and finite Lebesgue measure, /г(B) = X(A П B)/X(A) is the uniform probability measure on Bk n A.
