In 1933, A. Kolmogorov used H. Lebesgue's
measure theory to define random variables. With that definition, could have a random variable X whose values are only prime numbers and such that for each prime number p and for probability measure P,
P(X = p > 0),
that is, the probability that X = p is positive.
So, with X, the prime numbers are random.
References:
With TeX markup, polished details on measure theory are in
H.\ L.\ Royden, {\it Real Analysis: Second Edition,\/} Macmillan, New York, 1971.\ \
Walter Rudin, {\it Real and Complex Analysis,\/} ISBN 07-054232-5, McGraw-Hill, New York, 1966.\ \
and polished details on probability theory based on measure theory are in
Leo Breiman, {\it Probability,\/} ISBN 0-89871-296-3, SIAM, Philadelphia, 1992.\ \
Jacques Neveu, {\it Mathematical Foundations of the Calculus of Probability,\/} Holden-Day, San Francisco, 1965.\ \
P(X = p > 0),
that is, the probability that X = p is positive.
So, with X, the prime numbers are random.
References:
With TeX markup, polished details on measure theory are in
H.\ L.\ Royden, {\it Real Analysis: Second Edition,\/} Macmillan, New York, 1971.\ \
Walter Rudin, {\it Real and Complex Analysis,\/} ISBN 07-054232-5, McGraw-Hill, New York, 1966.\ \
and polished details on probability theory based on measure theory are in
Leo Breiman, {\it Probability,\/} ISBN 0-89871-296-3, SIAM, Philadelphia, 1992.\ \
Jacques Neveu, {\it Mathematical Foundations of the Calculus of Probability,\/} Holden-Day, San Francisco, 1965.\ \