ST213: Mathematics of Random Events
A list of things you should know!
 ALGEBRA: Definition of an algebra. Definition of the “smallest
algebra generated by a collection of sets”. How to construct the
smallest algebra generated by a given collection of sets. How to
check that a given collection of sets is an algebra. How to check
that a given collection is the smallest algebra containing a
collection of sets.
 -ALGEBRA: Definition of a -algebra. Definition of the
“smallest -algebra generated by a collection of sets”. How to
construct the smallest -algebra generated by a given collection of
sets, especially a collection of singletons. How to check that a
given collection of sets is a -algebra. How to prove that the algebra generated by two different collections of sets is the same.
-algebra. How to check that a given collection is the smallest algebra containing a collection of sets.
 LIMIT SETS: Know the definition of limit sets and what they
mean in terms of “how often an event happens”. Be able to write a
statement from analysis in set-theoretic terms. In particular, you
should know the definition of convergence of a sequence and you
should be able to write the set of points  for which a sequence
Xn() convergence in terms of limits sets.
 PROBABILITY MEASURES: You should know the definition
of a probability measure and how to check whether a given settheoretic function is a probability measure. You should know and
be able to prove the properties of a probability measure and use
them when required.
 EXTENSION THEOREM: You should be able to state
Caratheodory’s extension theorem. You should be able to use it in
order to decide whether the extension of a probability defined on
an algebra to a given -algebra is unique. If not, you should be able
to construct counter-examples, i.e. two probability measures that
are the same on the given algebra but different on the given algebra.
 RANDOM VARIABLES: You should know the definition of a
random variable/measurable function and be able to check whether
a given function is a random variable/measurable. You should
know the definition of the “smallest -algebra generated by a
random variable”.
 EXPECTATIONS: You should know the definition of an
indicator random variable and a simple random variable and the
definition of their expectation. You should know the definition of
the expectation for non-negative random variables and for arbitrary
random variables. You should know the Monotone Convergence
Theorem and the Dominated Convergence Theorem and be able to
use them in order to compute expectations of random variables that
are limits of a sequence of “simpler” random variables. You should
know, for example, how to use the Monotone Convergence
Theorem in order to define the expectation of a non-negative
random variable.
 BOREL-CANTELLI LEMMAS: You should be able to state and
prove both Borel-Cantelli Lemmas. You should be able to use the
Borel-Cantelli Lemmas in order to compute probabilities of limit
sets.
 CONVERGENCE OF SEQUENCES: You should know the
definition of “almost sure convergence” and “convergence in
probability”. You should be able to use the Borel-Cantelli Lemmas
to prove convergence of a sequence. You should be able to prove
Markov’s inequality and use it in order to prove convergence in
probability.
 LAW OF LARGE NUMBERS: You should be able to state and
apply the Law of Large Numbers.