# Integration

## Contents

# Integration#

(Simple function)

Let \(\mu\) be a measure on a set \(X\).
A *simple function* is any function of the form

where each \(a_i\in \mathbb R\) and the \(A_i\subset X\) are disjoint \(\mu\)-measurable sets. We treat \(0\cdot \infty=0\).

(Integral)

Let \(\mu\) be a measure on a set \(X\) and let \(f\colon X \to \mathbb R^+\). The (lower) integral of \(f\) with respect to \(\mu\) is

(Measurable function)

Let \(\mu\) be a measure on a set \(X\). A function \(f\colon X \to \mathbb R\) is \(\mu\)-measurable if \(f^{-1}((a,\infty))\) is \(\mu\)-measurable for every \(a\in \mathbb R\).

For \(f\colon X \to \mathbb R\) measurable, let \(f^+=\max\{f,0\}\) and \(f^-=\max\{-f,0\}\) (both \(\mu\)-measurable), so that \(f=f^+-f^-\) and \(|f|= f^+ + f^-\). If one of \(\int_X f^+\, \mathrm{d}\mu\) and \(\int_X f^-\, \mathrm{d}\mu\) are finite, we say that \(f\) is \(\mu\)-integrable and we define the integral of \(f\) with respect to \(\mu\) to be

If only \(\int_X f^-\, \mathrm{d}\mu <\infty\) (respectively \(\int_X f^+\, \mathrm{d}\mu <\infty\)) we write \(\int_X f\, \mathrm{d}\mu = \infty\) (respectively \(\int_X f\, \mathrm{d}\mu = -\infty\)).

Let \(X\) be a topological space. A function \(f\colon X\to \mathbb R\) is a Borel function if \(f^{-1}((a,\infty))\) is a Borel set for every \(a\in \mathbb R\).

There are some simple properties of the integral to check, such as linearity and monotonicity. See Example 14.

Linear combinations of measurable functions are measurable, as are limits of measurable functions. See Example 13.

(Fatou’s lemma)

Let \(\mu\) be a measure on a set \(X\) and \(f_k \colon X \to [0,\infty]\) \(\mu\)-measurable. Then

Proof. Let

be a simple function with

for each \(x\in A_i\) and each \(1\leq i \leq m\) and let \(0<t<1\). For each \(1\leq i \leq m\), the sets

monotonically increase to \(A_i\) as \(k\) increases. Therefore

and hence

Since \(0<t<1\) is arbitrary, the conclusion follows.

(Reverse Fatou)

Suppose that there exists \(g\geq 0\) with \(\int g\, \mathrm{d}\mu<\infty\) and \(f_k\leq g\) for all \(k\). Then

Indeed, this follows by applying Fatou’s lemma to \(g-f_k\).

(Monotone convergence theorem)

Let \(\mu\) be a measure on a set \(X\) and \(f_k\colon X \to [0,\infty]\) \(\mu\)-measurable. Suppose that for every \(x\in X\) and all \(k\in \mathbb N\), \(f_{k+1}(x) \geq f_k(x)\). Then

Proof. The monotonicity of the integral gives \(\leq\) whilst Fatou’s lemma gives \(\geq\).

(Dominated convergence theorem)

Let \(\mu\) be a measure on \(X\) and \(f_n\colon X \to \mathbb R\) \(\mu\)-measurable such that \(f_n\to f\) pointwise. Suppose that there exists measurable \(g\colon X \to [0,\infty]\) with \(\int g\, \mathrm{d}\mu<\infty\) such that \(|f_n(x)| \leq g(x)\) for all \(x\in X\). Then

Proof. Observe that for all \(n\in\mathbb N\), \(|f-f_n|\leq 2g\) and that \(\limsup |f-f_n| =0\). Then by the reverse Fatou lemma,

## Exercises#

For \(\mu\) a measure on a set \(X\), let \(f\colon X\to \mathbb R\) be measurable, respectively Borel. Show that the pre-image of any Borel \(B\subset \mathbb R\) is \(\mu\)-measurable, respectively Borel. Compare this to the definition of a continuous function.

Let \(\mu\) be a measure on \(X\) and for each \(i\in\mathbb N\) let \(f_i\colon X\to \mathbb R\) be \(\mu\)-measurable. Show that the functions

are \(\mu\)-measurable.

Show that a linear combination of \(\mu\)-measurable functions is \(\mu\)-measurable. Show that a countable (pointwise) sum of \(\mu\)-measurable functions is \(\mu\)-measurable.

There are some simple properties of the integral to check:

If \(f\leq g\) \(\mu\)-a.e. then

\[\int f d\mu \leq \int g\, \mathrm{d}\mu;\]The integral with respect to \(\mu\) is a linear operator;

If \(S\subset X\) is \(\mu\)-measurable then

\[\int_X f\, \mathrm{d}\mu = \int_S f \, \mathrm{d}\mu + \int_{X\setminus S} f\, \mathrm{d}\mu;\]\(|\int f\, \mathrm{d}\mu| \leq \int |f|\, \mathrm{d}\mu\);

etc…

Show that \(f\colon X \to \mathbb R\) is \(\mu\)-measurable if and only if

for every \(E\subset X\) and \(a<b\in \mathbb Q\).

State and prove a reverse monotone convergence theorem for monotonically decreasing sequences of functions.

Show that the Fatou lemma is false if the functions are not uniformly bounded below.

Show that the reverse Fatou lemma is false if the sequence is not bounded above by an integrable \(g\).

Show that the monotone convergence theorem is false if the sequence does not monotonically increase.

Let \(X,Y\) be sets, \(\mu\) a measure on \(X\) and \(f\colon X\to Y\). Show that \(f_{\#}\mu\) is a measure on \(Y\). If \(X,Y\) are topological spaces and \(\mu,f\) are Borel, show that \(f_{\#}\mu\) is a Borel measure on \(Y\).