8 1. Preparatory material

unaffected by modification on events of probability zero. Thus, a lack of

definedness on an event of probability zero will usually not cause diﬃculty,

so long as there are at most countably many such events in which one of

the probabilistic objects being studied is undefined. In such cases, one can

usually resolve such issues by setting a random variable to some arbitrary

value (e.g., 0) whenever it would otherwise be undefined.

We observe a few key subclasses and examples of random variables:

(i) Discrete random variables, in which R =

2R

is the discrete σ-

algebra, and R is at most countable. Typical examples of R include

a countable subset of the reals or complexes, such as the natural

numbers or integers. If R = {0, 1}, we say that the random variable

is Boolean, while if R is just a singleton set {c} we say that the

random variable is deterministic, and (by abuse of notation) we

identify this random variable with c itself. Note that a Boolean

random variable is nothing more than an indicator function I(E)

of an event E, where E is the event that the Boolean function

equals 1.

(ii) Real-valued random variables, in which R is the real line R and

R is the Borel σ-algebra, generated by the open sets of R. Thus

for any real-valued random variable X and any interval I, we have

the events “X ∈ I”. In particular, we have the upper tail event

“X ≥ λ” and lower tail event “X ≤ λ” for any threshold λ. (We

also consider the events “X λ” and “X λ” to be tail events;

in practice, there is very little distinction between the two types of

tail events.)

(iii) Complex random variables, whose range is the complex plane C

with the Borel σ-algebra. A typical event associated to a complex

random variable X is the small ball event “|X − z| r” for some

complex number z and some (small) radius r 0. We refer to

real and complex random variables collectively as scalar random

variables.

(iv) Given an R-valued random variable X, and a measurable map f :

R → R , the R -valued random variable f(X) is indeed a random

variable, and the operation of converting X to f(X) is preserved

under extension of the sample space and is thus probabilistic. This

variable f(X) can also be defined without reference to the sample

space as the unique random variable for which the identity

“f(X) ∈ S” = “X ∈ f

−1(S)”

holds for all R -measurable sets S.