Introduction to the Mathematical and Statistical Foundations of Econometrics

Borel Sets

An important special case of Definition 1.4 is where ^ = К and © is the collection of all open intervals:

© = {(a, b): Va < b, a, b є К}. (1.18)

Definition 1.5: The a-algebra generated by the collection (1.18) of all open intervals in К is called the Euclidean Borelfield, denoted by В and its members are called the Borel sets.

Note, however, thatB can be defined in different ways because the a - algebras generated by the collections of open intervals, closed intervals {[a, b] : Va < b, a, b є К} and half-open intervals {(-to, a] : Va є К}, respectively, are all the same! We show this for one case only:

Theorem 1.6: B = a ({(-to, a] : Va є К}).

Proof: Let

©+ = {(-to, a] : Va є К}. (1.19)

(a) If the collection © defined by (1.18) is contained in a (©*), then a (©*) is a a - algebra containing ©. ButB = a (©) is the smallest a-algebra containing ©; hence, B = a(©) c a(©*).

To prove this, construct an arbitrary set (a, b) in © out of countable unions or complements of sets in ©*, or both, as follows: Let A = (-to, a] and B = (-to, b], where a < b are arbitrary real numbers. Then A, B є ©+; hence, A, B є a(©*), and thus

~ (a, b] = (-to, a] U (b, to) = A U B є a(©*).

This implies that a(©*) contains all sets ofthe type (a, b]; hence, (a, b) = U^L^a, b - (b - a)/n] є a(©*). Thus, © c a(©*).

(b) If the collection ©* defined by (1.19) is contained in B = a (©), then a(©) is a a-algebra containing ©*. But a(©*) is the smallest a-algebra containing ©+; hence, a(©*) c a(©) = B.

To prove the latter, observe that, for m = 1, 2,..., Am = UTOLj(a - n, a + m-1) is a countable union of sets in ©; hence, Am є a(©), and consequently (-to, a] = nTOL Am = ~(UTO=j Am) є a(©). Thus, ©* c a (©) = B.

We have shown now that B = a(©) c a(©*) and a(©*) c a(©) = B. Thus, B and a(©*) are the same. Q. E.D.8

The notion of Borel set extends to higher dimensions as well:

See also Appendix 1.A.

Definition 1.6: B = а({хк=1(aj, bj): Waj < bj, aj, bj є К}) is the k - dimensional Euclidean Borel field. Its members are also called Borel sets (in

кк;.

Also, this is only one of the ways to define higher-dimensional Borel sets. In particular, as in Theorem 1.6 we have

Theorem 1.7: B = а({х^=1(-то, aj] : Waj є К}).

1.4. Properties of Probability Measures

The three axioms (1.8), (1.9), and (1.10) imply a variety of probability measure properties. Here we list only the most important ones.

Theorem 1.8: Let {^, P} be a probability space. The following hold for

sets in

(a) P(0) = 0,

(b) P(A) = 1 - P(A),

(c) A c B implies P(A) < P(B),

(d) P(A U B) + P(A П B) = P(A) + P(B),

(e) If An C An+1 for n = 1, 2,..., then P (An) t P (U^=1 An),

(f) If An D An+1 for n = 1, 2,..., then P (An) і P (П^ An),

(g) P(U~ 1 An) < 1 P(An).

Proof: (a)-(c): Easy exercises. (d) A U B = (A П B) U (A П B) U (B П A") is a union of disjoint sets; hence, by axiom (1.10), P(A U B) = P(A П

B) + P(A П B) + P(B П A). Moreover, A = (A П B) U (A П B) is a union

of disjoint sets; thus, P(A) = P(A П B) + P(A П B), and similarly, P(B) = P(B П A) + P(A П B). Combining these results, we find that part (d) fol­lows. (e) Let B1 = A1, Bn = An An-1 for n > 2. Then An = U”=1 Aj = Un=1 Bj and Uj= Aj = Uj=1 Bj. Because the Bj’s are disjoint, it follows from axiom (1.10) that

TO

U Aj) = E P(Bj)

n TO TO

= E P(Bj) + E P(Bj) = P(An) + E P(Bj)•

j=1 j=n+1 j=n+1

Part (e) follows now from the fact that^ffj=n+1 P (Bj) і 0. (f) This part follows from part (e) if one uses complements. (g) Exercise.

1.5. The Uniform Probability Measure

1.6.1. Introduction

Fill a bowl with ten balls numbered from 0 to 9. Draw a ball randomly from this bowl and write down the corresponding number as the first decimal digit of a number between 0 and 1. For example, if the first-drawn number is 4, then write down 0.4. Put the ball back in the bowl, and repeat this experiment. If, for example, the second ball corresponds to the number 9, then this number becomes the second decimal digit: 0.49. Repeating this experiment infinitely many times yields a random number between 0 and 1. Clearly, the sample space involved is the unit interval: ^ = [0, 1].

For a given number x є [0, 1] the probability that this random number is less than or equal to x is x. To see this, suppose that you only draw two balls and that x = 0.58. If the first ball has a number less than 5, it does not matter what the second number is. There are five ways to draw a first number less than or equal to 4 and 10 ways to draw the second number. Thus, there are 50 ways to draw a number with a first digit less than or equal to 4. There is only one way to draw a first number equal to 5 and 9 ways to draw a second number less than or equal to 8. Thus, the total number of ways we can generate a number less than or equal to 0.58 is 59, and the total number of ways we can draw two numbers with replacement is 100. Therefore, if we only draw two balls with replacement and use the numbers involved as the first and second decimal digit, the probability that we will obtain a number less than or equal to 0.58 is 0.59. Similarly, if we draw 10 balls with replacement, the probability that we will obtain a number less than or equal to 0.5831420385, for instance, is 0.5831420386. In the limit the difference between x and the corresponding probability disappears. Thus, for x є [0, 1] we have P([0, x]) = x. By the same argument it follows that for x є [0, 1], P({x}) = P([x, x]) = 0, that is, the probability that the random number involved will be exactly equal to a given numberx is zero. Therefore, for a givenx є [0, 1], P((0, x]) = P([0, x)) = P((0, x)) = x. More generally, for any interval in [0, 1] the corresponding probability is the length of the interval involved regardless of whether the endpoints are included. Thus, for 0 < a < b < 1, we have P([a, b]) = P((a, b]) = P([a, b)) = P((a, b)) = b — a. Any finite union of intervals can be written as a finite union of disjoint intervals by cutting out the overlap. Therefore, this probability measure extends to finite unions of intervals simply by adding up the lengths of the disjoint intervals in­volved. Moreover, observe that the collection of all finite unions of subintervals in [0,1], including [0, 1] itself and the empty set, is closed under the formation of complements and finite unions. Thus, we have derived the probability mea­sure P corresponding to the statistical experiment under review for an algebra ^0 of subsets of [0, 1], namely,

^0 = {(a, b), [a, b], (a, b], [a, b), Va, b є [0, 1], a < b,

and their finite unions}, (120)

where [a, a] is the singleton {a} and each of the sets (a, a), (a, a] and [a, a) should be interpreted as the empty set 0. This probability measure is a special case of the Lebesgue measure, which assigns each interval its length.

If you are only interested in making probability statements about the sets in the algebra (1.20), then you are done. However, although the algebra (1.20) contains a large number of sets, we cannot yet make probability statements involving arbitrary Borel sets in [0, 1] because not all the Borel sets in [0, 1] are included in (1.20). In particular, for a countable sequence of sets Aj є ^o, the probability P(UJL1 Aj) is not always defined because there is no guarantee that U=j Aj є ^o. Therefore, to make probability statements about arbitrary Borel set in [0, 1], you need to extend the probability measure P on to a probability measure defined on the Borel sets in [0, 1]. The standard approach to do this is to use the outer measure.