1 Introduction

1.1 Catch-up Prerequisites

Basic Topological spaces and analysis. Definitions of basic concepts including continuity, convergence, topo-
logical spaces, it would be sensible to know what Riemann integration. Basic linear algebra, what a vector
space is and what a norm is. I will briefly mention Hilbert spaces and Banach spaces but not introduce
them.

2 Measure Spaces
2.1 What is a measure space?
We begin with a formal definition of what a measure space is. We have a triple (E, E, µ) here E is some set.
It is the place that we want to ‘measure’ parts of. It could be Rn or some space of possible outcomes of an
experiment like coin tossing. E is called the σ-algebra. A σ-algebra is a set of subsets of E. These will be
the sets which we can measure. The sets in a σ-algebra must obey a set of rules which are similar to those
obeyed by the open or closed sets in a topology.
(i) ∅, E ∈ E,
(ii) A ∈ E ⇒ Ac ∈ E. S
(ii) {An , n ∈ N} ⊂ E ⇒ n An ∈ E. An easy way to think about this is that a σ-algebra is closed under
taking complements and countable unions or intersections. The very easiest example of a pair (E, E) to think
about is if E is some finite set of points for example E = {1, 2, 3}. and E is all the possible subsets. We say
a subset, A, of E generates the σ-algebra if E is the smallest σ-algebra containing A.
The final element µ is the measure. This tells us how big each of the sets are. Fomally µ is a function
from E to [0, ∞]. It also has to follow a set of rules.
(i) µ(∅)S= 0. P
(ii) µ ( n An ) = n µ(An ) when the An are all pairwise disjoint.
This last property is called countable additivity.

2.2 Borel σ-algebras

Apart from countable state spaces the most common explicit σ-algebra which appears in applications in the
Borel σ-algebra. This is generated when we have a topological space E, and we define the Borel σ-algebra
B to be the smallest σ algebra which contains all the open sets of E.

2.3 π-systems and d-systems

It is useful in several proofs and applications of measure theory to look at other structure on subsets of E.
So here we quickly give the definitions.
- A π-system, A, is a set containing ∅ with the property that if A, B ∈ A then A ∩ B ∈ A.
- A d-system, D, is a set containing ES with the property that if A, B ∈ D, A ⊂ B then we have B A ∈ D
and that if A1 ⊂ A2 ⊂ A3 · · · ⊂ D then n An ∈ D.
A very useful theorem in measure theory is
Theorem 1. If we have two measures µ1 , µ2 , on a measurable space (E, E) and there exists A, a π-system
generating E on which µ1 and µ2 agree then µ1 = µ2 .

2.4 Lebesgue Measure

Lebesgue measure is probably the most famous and fundamental measure. All the details of its construction
would take too long. It is a measure on Rn and is that which corresponds to our intuitive idea of how bit a

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set is. In 1-D it is defined by
µ((a, b]) = b − a,
where (a, b] is any interval. This property completely determines Lebesgue measure on the Borel σ-algebra
of R.

2.5 Measure Theoretic Formulation of Probability

In Probability theory it is common to write the measure space as (Ω, F , P) with the additional assumption
that P(Ω) = 1. In this setting Ω is the set of all possible individual outcomes (of the experiment, or random
process). For an example suppose we are going to toss a coin twice and we want to look at what results we
get. Our set is that of all possible sequences

Ω = {T T, T H, HT, HH}.

We can simply define the σ-algebra to be PΩ. Then we can see that if we want to see the probability of
something occuring, for instance at least one head appearing, then this defines a subset of Ω which is in F
and we can find this probability. P( At least one head appears ) = P({T H, HT, HH}).

2.6 Exercises
Here are some hopefully straightforwardTexercises:
1. Prove that if (An , n ∈ N) ⊂ E ⇒ n An ∈ E.
2. Prove that if E is a countable set then PE is a σ-algebra.S P
3. Is it always the case thatTif all the An are in E then µ ( n An ) ≤ T n µ(An ).
4. If An are all in E and µ ( n An ) = 0 is it necessarilly the case that n An = ∅.
5. If E is both a π-system and a d-system prove that it is a σ-algebra.
6. Speculate on how Lebesgue measure is defined in higher dimensions.

3 Functions
3.1 Measurable Functions
Having our measure spaces allows us to start looking at functions on measure spaces. That is, if (E1 , E1 )
and (E2 , E2 ) are measureable spaces we can look at functions

f : E1 → E2 .

We would like to restric to a set of functions which works with our σ-algebra so we define a measurable
function to be s.t.
A ∈ E2 ⇒ f −1 (A) ∈ E1 .
This is analagous to the continuity definition for topological spaces.
There is a useful theorem for finding out the properties of measureable functions.
Theorem 2 (Monotone Class Theorem). Let (E, E) be a measurable space and let A be a π-system generating
E. Let V be a vector space of bounded functions f : E → R then if
1. 1 ∈ V and 1A ∈ V for every A ∈ A.
2. If fn is a sequence of functions in V with fn ↑ f for some bounded functions f then f ∈ V.
Then V will contain all the bounded measurable functions.

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3.2 Random Variables
A random variable is a measurable function from a probability space. For example if we again have the
probability space generated by tossing a coin twice. Then if X counts the number of heads, it is a random
variable with landing space N with σ-algebra PN often the landing space of a random variable is not made
specific. In particular its σ algebra may not be made explicit. The random variable induces a measure on
its landing space
µx = P ◦ X −1 .
This is called the law of X.

3.3 Convergence of Measurable Functions.

There are two types of convergence for measurable functions which are different from those for normal
functions. There is an additional one for random variables. They are
Convergence almost everywhere this means that if fn → f almost everywhere (a.e. or almost surely on
a probability space) then the set N on which fn (x) does not converge to f (x) has µ(N ) = 0. In general
almost everywhere or almost surely is used to refer to a property that holds everywhere except a set which
has measure 0.
Convergence in measure fn converges to f in probability if µ({x : |fn (x) − f (x)| > }) → 0 as n → ∞
for each . This is also called convergence in probability on a probability space.
These two notions are close to being equivalent (at least on finite measure spaces). This is made clear
by the following theorem.
Theorem 3. Let fn be a sequence of measurable functions on a measure space (E, E, µ).
1. If µ(E) < ∞ (the space is finite) then convergence a.e. implies convergence in measure.
2. If the fn converge in probability then there exists a subsequence fnk which converges a.e.
Because probability spaces are always finite we can say convergence a.s. implies convergence in probability.
The last type of convergence is convergence in distribution. The distribution function of a random variable
is FX = P(X ≤ x). Then we say Xn → X in distribution if FXn (x) → FX (x) for every x which is a continuity
point of FX . This differes from the others as the Xn , X do not need to be defined on the same probability
space. Again this is related to the other forms of convergence.
Theorem 4. Let Xn be a sequence of random variable and X another random variable. If they are all
defined on the same probability space the convergence in probability implies convergence in distribution. If
Xn → X in distribution then there exist other random variables X̃n , X̃ defined on the same probability space
such that Xn → X a.s.

3.4 Exercises
Exercises on functions:
1. Prove that it is only necessary to check the measurablity criterion on a π-system generating the
σ-algebra.
2. Prove that a continuous function between two topological spaces with their Borel σ-algebras is mea-
surable.
3. Prove that if f , g are measureable functions into R with its Borel σ-algebra then f g and f + g are
also measurable.
4. Prove that if fn are all measurable functions into R with its Borel σ-algebra then inf n fn , supn fn ,
lim inf n fn and lim supn fn are all measurable.
5. Try and come up with sequences that converge either in measure or probability but not in both.

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4 Integration
The theoryRof Lebesgue integration
R allows you to integrate functions with respect to a measure. This is
written as E f dµ, µ(f ) or E f (x)µ(dx). I wont go into the detail of how the integral is constructed or
shown to exist as it is not needed for most applications. The notaion µ-integrable is used of a function when
µ(|f |) exists and is finite.

4.1 Convergence of Integrals

There are three main convergence theorems for integrals.
Theorem 5 (Monotone Convergence). Let fn be a sequence of positive functions such that fn ↑ f then
µ(fn ) ↑ µ(f ).

Theorem 6 (Fatou’s Lemma). Let fn be a sequence of positive functions then µ(lim inf n fn ) ≤ lim inf n µ(fn ).
Theorem 7. Let fn be a sequence of measurable functions and fn (x) → f (x) and that |f (x)| ≤ g(x) for
every x and µ(g) < ∞. Then f, fn are integrable and µ(fn ) → µ(f ). We call g the dominating function.

4.2 Product Measure and Fubini’s Theorem

If (E1 , E1 , µ1 ) and (E2 , E2 , µ2 ) are probability space then we can construct the probability space (E1 ×E2 , E1 ×
E2 , µ1 × µ2 ). We do this by looking at the π-system of sets of the form A × B where A ∈ E1 and B ∈ E2 .
Then we can define E1 × E2 to be the σ-algebra generated by this π-system. We can also extend a measure
defined on this π system by µ1 × µ2 (A × B) = µ1 (A)µ2 (B). Fubini’s theorem says that if f : E1 × E2 is a
E1 × E2 measurable function then
Z Z 
µ1 × µ2 (f ) = f (x1 , x2 )µ2 (dx2 ) µ1 (dx1 ).
E1 E2

4.3 Lebesgue Spaces

An important class of function spaces are Lebesgue spaces of Lp spaces. Essentially for 1 ≤ p < ∞ these are
spaces of functions from E to R such that µ(|f |p ) < ∞ equipped with the norm kf kp = µ(|f |p )1/p . However
this is not a true norm since there are many functions with f = 0 a.e. therefore we have to consider instead
of f the equivalence class of functions {g : g = f a.e.}. This complication is often suppressed and you don’t
really notice it. L∞ is the space of functions whose essential supremum is finite. The essential supremum is
defined by
kf k∞ = inf sup |f (x)|.
N ∈N x∈E−N

Where N is the collection of sets in E where µ(N ) = 0 (null sets). The Lp spaces are Banach spaces with
this norm and L2 is a Hilbert space with the inner product
Z
< f, g >= f gdµ.
E

4.4 Exercises
1. Find a sequence of measurable functions fn such that fn converges to some function f with µ(|f |) < ∞,
but µ(fn ) doesn’t converge to µ(f ).
2. The sequence
2n r
X k
fn (x) = 1[(k−1)/2n ,k/2n )
2n
k=1

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converges to f (x). Prove that µ(fn ) converges to µ(f ) where µ is Lebesgue measure on [0, 1].
3. Check that the π-system defined in the making of the product measure space is in fact a π-system.
4. Find a function which is in L2 (R) but not in L1 (R) and vice versa.

5 Inequalities
In both probability and PDEs common inequalities appear very frequently and may not always be referenced
directly. Here is a list of what I think are the most common and useful.
Theorem 8 (Markov’s Inequality). This is sometimes also called Chebychev’s inequality but that is also a
name given to a corollary of Markov’s inequality. This says that if X is a random variable and λ > 0 then
E(X)
P(X ≥ λ) ≤ .
λ
Theorem 9 (Jensen’s Inequality). Suppose we have φ is a convex function (for t ∈ (0, 1), φ(tx + (1 − t)y) ≤
tφ(x) + (1 − t)φ(y)) and (E, E, µ), a measure space with µ(E) < ∞. Then we will have

µ(φ(f )) ≤ φ(µ(f )).

Theorem 10 (Holder’s Inequality). If we have 1/p + 1/q = 1 and f ∈ Lp , g ∈ Lq then we have

kf gk1 ≤ kf kp kgkq .

In the case where p = q = 2 this is called Cauchy-Schwarz inequality.

Theorem 11 (Minkowski’s Inequality). This one is easy to remember because it is the triangle inequality
for Lp norms, but it is not so easy to prove. If f, g are in Lp then

kf + gkp ≤ kf kp + kgkp

Theorem 12 (Young’s Inequality). There is a more complicated inequality of young involving convolutions.
This inequality is not about measure theory but it is very useful particularly in PDEs after using Cauchy-
Schwarz. If a, b are real numbers
1
|ab| ≤ (a2 + b2 ),
2
and following this
 1
|ab| ≤ a2 + b2 .
2 2

5.1 Exercises
1. Show if X is a Normal random variable with mean 0 and variance 1. Show that
1
P(X ≥ α) ≤ .
2α2
2. Show that if f, g ∈ L2 (µ) then
Z Z Z
1 1
| f (x)g(x)µ(dx)| ≤ |f (x)|2 µ(dx) + |g(x)|2 µ(dx).
2 2
3. Show that if f ∈ L1 ∩ L2 then
!2
Z T Z T
2 1
f (x) dx ≤ f (x)dx .
0 T 0

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6 Useful Things I have skipped
These are mainly related to probability and can be found in the probability and measure notes.
1. Borel-Cantelli Lemmas,
2. Kolmogorov 0-1 Law,
3. Fourier Theory Relating to L2 .
4. Uniform Integrability