Celebratio Mathematica

[1]D. Blackwell: “On a theorem of Lyapunov,” Ann. Math. Stat. 22 : 1 (March 1951), pp. 112–114. MR 0039033 Zbl 0042.28502 article

[2]R. Bellman and D. Blackwell: “On moment spaces,” Ann. Math. (2) 54 : 2 (September 1951), pp. 272–274. MR 0043866 Zbl 0044.12601 article

[3]D. Blackwell: “The range of certain vector integrals,” Proc. Am. Math. Soc. 2 : 3 (September 1951), pp. 390–395. MR 0041195 Zbl 0044.27702 article

[4]D. Blackwell: “A representation problem,” Proc. Am. Math. Soc. 5 : 2 (1954), pp. 283–287. MR 0061653 Zbl 0055.28804 article

The problem solved in this paper is the following. For a fixed number \( a \), \( 0 < a < 1 \), what functions \( f(x) \) on \( 0 \leq x \leq 1 \) have a representation \begin{equation*}\tag{1} f(x) = \sum_1^{\infty}c_n\varphi_n(x), \end{equation*} where \( c_n\geq 0 \), \( \sum_1^{\infty}c_n \) converges, and each \( \varphi_n \) is the characteristic function of a subset of \( 0\leq x \leq 1 \) of Lebesgue measure \( a \)? Clearly any \( f \) satisfying (1) satisfies \begin{equation*}\tag{2} 0\leq f(x)\leq\frac{1}{a}\int_0^1 f(x)\,dx \quad\text{for all }x, \end{equation*} since \[ f(x)\leq\sum_1^{\infty}c_n=\frac{1}{a}\int_0^1 f(x)\,dx .\] The result of this paper is that (2) is sufficient as well as necessary for a function \( f \) to admit a representation (1).

[5]D. Blackwell: “On discrete variables whose sum is absolutely continuous,” Ann. Math. Stat. 28 : 2 (1957), pp. 520–521. MR 0088091 Zbl 0078.31602 article

If \( \{Z_n\} \), \( n = 1, 2, \ldots \) is a stationary stochastic process with \( D \) states \[ 0, 1, \dots, D - 1 \quad\text{and}\quad X = \sum^\infty_1 \frac{Z_k}{D^n} ,\] Harris [1955] has shown that the distribution of \( X \) is absolutely continuous if and only if the \( Z_n \) are independent and uniformly distributed over \( {}0 \), \( 1, \dots \), \( D - 1 \), i.e., if and only if the distribution of \( X \) is uniform on the unit interval. In this note we show that if \( \{Z_n\} \), \( n = 1 \), \( 2, \dots \) is any stochastic process with \( D \) states \( {}0 \), \( 1, \dots \), \( D - 1 \) such that \[ X = \sum^\infty_1 \frac{Z_n}{D^n} \] has an absolutely continuous distribution, then the conditional distribution of \[ R_k = \sum^\infty_{n = 1} \frac{Z_{k + n}}{D^n} \] given \( Z_1, \dots \), \( Z_k \) converges to the uniform distribution on the unit interval with probability 1 as \( k \to \infty \). It follows that the unconditional distribution of \( R_k \) converges to the uniform distribution as \( k \to \infty \). Since if \( \{Z_n\} \) is stationary the distribution of \( R_k \) is independent of \( k \), the result of Harris follows.

[6]D. Blackwell and L. E. Dubins: “A converse to the dominated convergence theorem,” Illinois J. Math. 7 : 3 (1963), pp. 508–514. MR 0151572 Zbl 0146.37503 article

On a probability space \( (\Omega, \mathcal{B}, P) \), let \( \{f_n \), \( n=1 \), \( 2,\dots\} \) be a sequence of nonnegative random variables in \( L_1 \) such that \( f_n \to f\in L_1 \) with probability 1 and define \( g=\sup_n f_n \). If \( g\in L_1 \), the Lebesgue dominated convergence theorem asserts that \( E(f_n)\to E(f) \). More generally, as noted by [Doob 1953, p. 23], if \( g\in L_1 \), then for any Borel field \( \mathcal{B}_0 \) contained in \( \mathcal{B} \), \[ E(f_n\mid\mathcal{B}_0) \to E(f\mid\mathcal{B}_0) \quad\text{a.e.} \] If one extends this result in a minor manner, Lebesgue’s condition \( g\in L_1 \) is not only sufficient but necessary, as the following converse to the dominated convergence theorem asserts.

If \( f_n \geq 0 \), \( f_n \to f \) a.e., \( f_n\in L_1 \), and \( g = \sup_n f_n \notin L_1 \), there are, on a suitable probability space, random variables \( \{f_n^* \), \( n=1 \), \( 2,\dots\} \), \( f^* \), and a Borel field \( \mathcal{C} \) such that \( f^* \), \( f_1^* \), \( f_2^*,\dots \) have the same joint distribution as \( f \), \( f_1 \), \( f_2,\dots \), and \[ P\bigl\{E(f_n^*\mid\mathcal{C})\to E(f^*\mid\mathcal{C})\bigr\} = 0. \]

[7]D. Blackwell and L. E. Dubins: “Sharp bounds on the distribution of the Hardy–Littlewood maximal function,” Proc. Am. Math. Soc. 14 : 3 (1963), pp. 450–453. MR 0148842 Zbl 0118.05401 article

[8]D. Blackwell: “A Borel set not containing a graph,” Ann. Math. Stat. 39 : 4 (1968), pp. 1345–1347. MR 0229451 Zbl 0177.48401 article

Examples of Borel sets \( X, Y, B \) such that

\( B \subset X \times Y \),
the projection of \( B \) on \( X \) is \( X \), but
for no Borel-measurable \( d \) mapping \( X \) into \( Y \) is the graph of \( d \) a subset of \( B \),

have been given by [Novikoff 1931; Sierpiński 1931; Addison 1958]. Such examples are of interest in dynamic programming (see for instance [Blackwell 1965]), since if we interpret \( X \) as the set of states of some system, \( Y \) as the set of available acts, and \( I_B(x,y) \) where \( I_B \) is the indicator of \( B \), as your income if the system is in state \( x \) and you choose act \( y \), you can earn 1 in every state, but there is no Borel measurable plan, i.e. function \( d \) from \( X \) into \( Y \), with \( d(x) \) specifying the act to be chosen when the system is in state \( x \), that earns 1 in every state.

This note presents a new example \( X \), \( Y \), \( B \), simpler than those previously given. The proof that it is an example uses ideas from Addison’s construction, and a theorem of Gale and Stewart [1953] on infinite games of perfect information, and is somewhat more complicated than Addison’s

[9]D. Blackwell: “Borel-programmable functions,” Ann. Probab. 6 : 2 (1978), pp. 321–324. MR 0460573 Zbl 0398.28002 article

[10]D. Blackwell: “There are no Borel SPLIFs,” Ann. Probab. 8 : 6 (1980), pp. 1189–1190. MR 602393 Zbl 0451.28001 article

[11]D. Blackwell and A. Maitra: “Factorization of probability measures and absolutely measurable sets,” Proc. Am. Math. Soc. 92 : 2 (1984), pp. 251–254. MR 754713 Zbl 0554.60001 article

[12]D. Blackwell and P. Diaconis: “A non-measurable tail set,” pp. 1–5 in Statistics, probability and game theory: Papers in honor of David Blackwell. Edited by T. S. Ferguson, L. S. Shapley, and J. B. MacQueen. IMS Lecture Notes–Monograph Series 30. Institute of Mathematical Statistics (Hayward, CA), 1996. MR 1481768 incollection

Lloyd Stowell Shapley	Related
Thomas S. Ferguson	Related
James B. MacQueen	Related
Persi Diaconis	Related

David H. Blackwell

Measure & integration