#### by Murray Rosenblatt

I had only occasional contact with David Blackwell through the years. But I always found him to be a warm, gracious person with a friendly greeting. He entered the University of Illinois at Urbana–Champaign in 1935 at the age of sixteen and received a bachelor’s degree in mathematics in 1938 and a master’s degree in 1939. Blackwell wrote a doctoral thesis on Markov chains with Joseph L. Doob as advisor in 1941. Two earlier almost-contemporary doctoral students of J. L. Doob were Paul Halmos, with a doctoral degree in 1938, and Warren Ambrose, with the degree in 1939.

Blackwell was a postdoctoral fellow at the Institute for Advanced Study for a year from 1941 (having been awarded a Rosenwald fellowship). There was an attempted racist intervention by the then-president of Princeton, who objected to the honorific designation of Blackwell as a visiting fellow at Princeton (all members of the Institute had this designation). He was on the faculty of Howard University in the mathematics department from 1944 to 1954. Neyman supported the appointment of David Blackwell at the University of California, Berkeley, in 1942, but this fell through due to the prejudices at that time (see [e4]). However, in 1955 David Blackwell was appointed professor of statistics at UC Berkeley and became chair of the department the following year.

Blackwell wrote over ninety papers and made major contributions in many areas — dynamic programming, game theory, measure theory, probability theory, information theory, and mathematical statistics. He was an engaging person with broad-ranging interests and deep insights. He was quite independent but often carried out research with others. Interaction with Girshick probably led him to research on statistical problems of note. Researches with K. Arrow, R. Bellman, and E. Barankin focused on game theory. Joint work with A. Thomasian (a student of his) and L. Breiman was on coding problems in information theory. He also carried out researches with colleagues at UC Berkeley, such as David Freedman, Lester Dubins, J. L. Hodges, and Peter Bickel. The Rao–Blackwell theorem dealing with the question of optimal unbiased estimation is due to him.

He was elected the first African American member of the National Academy of Sciences, USA, and received many other awards. He was a distinguished lecturer. We’re thankful that he survived the difficulties that African Americans had to endure in a time of great bias (in his youth). He was a person of singular talent in the areas of statistics and mathematics.

I shall describe limited aspects of the research
of Blackwell and Dubins
[e2]
on regular conditional
distributions (see Doob
[e1]
for a discussion of
conditional probability). This was an area that
Blackwell often found of interest. Given a measurable
space __\( (\Omega, \mathcal{B}) \)__ with __\( \mathcal{A} \)__ a sub __\( \sigma \)__-field of the
__\( \sigma \)__-field __\( \mathcal{B} \)__, call __\( P \)__ defined on __\( \Omega \times \mathcal{B} \)__ a *regular conditional
distribution* (r.c.d.) **\textit**{for __\( \mathcal{A} \)__ on __\( \mathcal{B} \)__} if for all
__\( \omega \in \Omega \)__, __\( B \in \mathcal{B} \)__,

__\( P(\omega, \cdot) \)__is a probability measure on__\( \mathcal{B} \)__.For each

__\( B \in \mathcal{B} \)__,__\( P(\cdot, B) \)__is__\( \mathcal{A} \)__-measurable and related to the probability distribution via__\[ \int_A P(\omega, B)\,dP(\omega) = P(A \cap B) \]__for__\( A \in \mathcal{A} \)__,__\( B \in \mathcal{B} \)__.

Such regular conditional distributions do not always
exist. But assuming existence, call it *proper*
if
__\[
P(\omega, A) = 1
\]__
whenever __\( \omega \in A \in \mathcal{A} \)__.

The probability measure __\( P \)__ on __\( \mathcal{A} \)__ is called
extreme if __\( P(A) = 0 \)__ or 1 for all __\( A \in \mathcal{A} \)__. An __\( \mathcal{A} \)__-atom
is the intersection of all elements of __\( \mathcal{A} \)__ that contain
a given point of __\( \Omega \)__. If for __\( A \in \mathcal{A},\, P(A) = 1 \)__, __\( P \)__ is
said to be *supported* by __\( A \)__.

Then we have:

Assume __\( \mathcal{B} \)__ is countably generated. Then
each of the conditions implies the successor.

There is an extreme countably additive probability measure on

__\( \mathcal{A} \)__that is supported by no__\( \mathcal{A} \)__-atom belonging to__\( \mathcal{A} \)__.__\( \mathcal{A} \)__is not countably generated.No regular conditional distribution for

__\( \mathcal{A} \)__on__\( \mathcal{B} \)__is proper.

This result shows that, for __\( \Omega \)__ the infinite product
of a separable metric space containing more
than one point, neither the tail field, the field of
symmetric events, nor the invariant field admit
a proper r.c.d (regular conditional distribution).
They weaken the countable additivity condition of
an r.c.d. to finite additivity and add (1) to obtain
the notion of a *normal conditional distribution* and
arrive at sufficient conditions for existence. Later
related research by
Berti
and
Rigo
[e3]
considers the
r.c.d.s with appropriate weakenings of the concept
of proper.