Definition 1.3. Let (Ω, F) and (Ω
0
, F
0
) be two measurable spaces. We say a
function f : (Ω, F) → (Ω
0
, F
0
) is measurable if it respects the σ-algebra struc-
ture. In other words, for any B ∈ F
0
, we have that f
−1
(B) ∈ F. If F = σ(X),
then it is suﬃcient to check this for any subset of X instead. From now on, we
write f : Ω → Ω
0
instead of the more cumbersome notation above.
This makes clearer the notion of ”measurability” and how similar it is to
topology. In topology, open sets specify which functions are continuous. In
measurable spaces, elements of σ-algebra specify which functions are measur-
able.
Proposition 1.4. Measurable functions respect the usual algebraic operations.
Sums, (scalar and pointwise) products, quotients and compositions of measurable
functions are measurable where it makes sense. Furthermore, if the target space
of the function is a metric space, pointwise limits of measurable functions are
measurable. Supremums, inﬁmums and their limits of real-valued measurable
functions are measurable as well.
Proof. The proofs are either not very illuminating or extremely easy. We just
show composition. Suppose f : Ω
1
→ Ω
2
and g : Ω
2
→ Ω
3
are measurable
functions whose domains and codomains match as one would want. Then the
preimage of g ◦ f is f
−1
(g
−1
(A)). Suppose A ∈ F
3
. Then B = g
−1
(A) ∈ F
2
.
In turn, C = f
−1
(B) ∈ F
1
. This holds because both f and g are measurable.
Thus for A ∈ F
3
, (g ◦f) (A) ∈ F
1
which shows g ◦ f is measurable.
As one can see, the class of measurable functions is quite rich. This is a good
thing. Since we will deﬁne integration on those functions, we want as many
functions as possible to be at least measurable. It is obvious that continuous
functions are measurable provided that the σ-algebra on the domain is the one
generated by open (or closed) sets. We will call this σ-algebra the Borel algebra.
In the case of the real line, we will write it as B(R). We now deﬁne the main
object of study in measure theory.
Definition 1.5. A measure is a function µ : F → R
+
∪ {∞} such that the
following holds:
1. µ(∅) = 0
2. µ(
S
∞
i=1
X
i
) =
P
∞
i=1
µ(X
i
), for any countable family of disjoint sets (X
i
)
i∈I
.
The second property is called σ-additivity (sometimes also known as countable
additivity). If µ(Ω) = 1, we say that µ is a probability measure, or more simply,
a probability. The triple (Ω, F, µ) is called a measure space. If µ = P is a
probability, then we call the space (Ω, F, P) a probability space, and a measurable
function on it a random variable.
Proposition 1.6 (Properties of measures). Let (Ω, F, µ) be a measure space.
Let A, B ∈ F. Then:
1. µ(A) = µ(A \ B) + µ(A ∩ B);
4