Integration#

Definition 10 (Simple function)

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

\[\sum_{i=1}^m a_i \chi_{A_i},\]

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

Definition 11 (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

\[\int f\, \mathrm{d}\mu:= \sup\left\{\sum_{i=1}^m a_i \mu(A_i) : s=\sum_{i=1}^m a_i \chi_{A_i}\leq f,\ s \text{ simple}\right\}.\]

Definition 12 (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

\[\int_X f\, \mathrm{d}\mu = \int_X f^+ \, \mathrm{d}\mu - \int_X f^- \, \mathrm{d}\mu.\]

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.

Theorem 3 (Fatou’s lemma)

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

\[\int_X \liminf_{k\to\infty} f_k \, \mathrm{d}\mu \leq \liminf_{k\to\infty} \int_X f_k\, \mathrm{d}\mu.\]

Proof. Let

\[s=\sum_{i=1}^m a_i \chi_{A_i}\]

be a simple function with

\[s\leq \liminf f_k.\]

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

\[G_{k,i}:= \{x\in A_i: f_k(x)\geq t a_i \text{ for all } j\geq k\}\]

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

\[\int f_k \, \mathrm{d}\mu \geq \sum_{i=1}^m t a_i \mu(G_{k,i}) \to \sum_{i=1}^n t a_i \mu(A_{i}).\]

and hence

\[\sum_{i=1}^n t a_i \mu(A_{i}) \leq \liminf_{k\to\infty} \int f_k\, \mathrm{d}\mu.\]

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

Remark 3 (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

\[\limsup_{k\to\infty} f_k\, \mathrm{d}\mu \geq \limsup_{k\to\infty} \int_X f_k \, \mathrm{d}\mu.\]

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

Theorem 4 (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

\[\lim_{k\to\infty}\int f_k \, \mathrm{d}\mu = \int \lim_{k\to\infty}f_k\, \mathrm{d}\mu.\]

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

Theorem 5 (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

\[\int f_n \, \mathrm{d}\mu \to \int f\, \mathrm{d}\mu.\]

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,

\[\left|\int f\, \mathrm{d}\mu -\int f_n \, \mathrm{d}\mu\right| \leq \int |f-f_n|\, \mathrm{d}\mu \to 0.\]

Exercises#

Example 12

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.

Example 13

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

\[\limsup_{i\to\infty} f_i \quad \text{and} \quad \liminf_{i\to\infty}f_i\]

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.

Example 14

There are some simple properties of the integral to check:

  1. If \(f\leq g\) \(\mu\)-a.e. then

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

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

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

  5. etc…

Example 15

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

\[\mu(E) \geq \mu(E \cap f^{-1}((-\infty,a))) + \mu(E\cap f^{-1}((b,\infty)))\]

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

Example 16

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

Example 17

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.

Example 18

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 \(Y\).