Probability with Measure

4.9 Multiple integrals and function spaces \((\star )\)

This section is included for interest. It is marked with a \((\star )\) and it is off-syllabus. It includes two separate topics.

4.9.1 Fubini’s Theorem \((\star )\)

Fubini and Tonelli’s theorems let us deal with expressions involving multiple integrals, for example of the form \(\int _{S_1}\int _{S_2}f(x,y)\,dx\,dy\). They give different sets of conditions under which we can change the order of integration. Loosely, Tonelli’s theorem is an analogue of the MCT and Fubini’s theorem is an analogue of the DCT. Before introducing them we need to handle some measure theoretic details.

For \(i=1,2\) let \((S_i,\Sigma _i,m_i)\) be measure spaces and recall the product measure space \((S_{1} \times S_{2}, \Sigma _{1} \otimes \Sigma _{2}, m_{1} \times m_{2})\) introduced in Section 1.9. If \(f:S_1\times S_2\to \R \) is measurable then the coordinate projections \(x\mapsto f(x,y)\) and \(y\mapsto f(x,y)\) are measurable for almost all \(x\in S_1\) and \(y\in S_2\). Moreover, if \(f\in \Lone (S_1\times S_2)\) then these coordinate projections are respectively in \(\mc {L}^(S_1)\) and \(\Lone (S_2)\), again for almost all \(x\in S_1\) and \(y\in S_2\), and the same is true of the functions \(x \rightarrow \int _{S_{2}}f(x,y)m_{2}(dy)\) and \(y \rightarrow \int _{S_{2}}f(x,y)m_{2}(dx)\). We won’t include a proof of these claims here. They put us a in a position to state the key result of this section.

Theorem 4.9.1 Let \(f:S_1\times S_2\to \R \) be measurable. Suppose that at least one of the following conditions holds.
- 1. Fubini’s Theorem: \(f\in \Lone (S_1\times S_2)\).
- 2. Tonelli’s Theorem: \(f\geq 0\).
Then
\(\seteqnumber{0}{4.}{23}\)
\begin{align*} \int _{S_{1} \times S_{2}}f \,d(m_{1} \times m_{2}) & = \int _{S_{1}}\left (\int _{S_{2}}f(x,y)\,dm_{2}(y)\right )dm_1(x)\\ & = \int _{S_{2}}\left (\int _{S_{1}}f(x,y)\,dm_{1}(x)\right )\,dm_2(y). \end{align*}

In the case of Fubini’s theorem \(\int _{S_{1} \times S_{2}}f \,d(m_{1} \times m_{2})\) is a real number, whilst in the case of Tonelli’s theorem it is in \([0,\infty ]\). These results are named after the Italian mathematicians Guido Fubini (1879-1943) and Leonida Tonelli (1885-1946). They are very important results, equal in stature to the MCT and DCT, but we omit a full treatment of them from our course in order to progress on to thinking about probability.

4.9.2 Function Spaces \((\star )\)

This section is aimed at those taking courses in functional analysis. An important application of Lebesgue integration is to the construction of Banach spaces \(L^p(S, \Sigma , m)\) of equivalence classes of real-valued functions, where the equivalence relation \(f\eqae g\) is used on \(\mc {L}^p(S,\Sigma ,m)\), and which satisfy the requirement

\[ ||f||_{p} = \left (\int _{S}|f|^{p}\,dm\right )^{\frac {1}{p}} < \infty ,\]

where \(1 \leq p < \infty \). The function \(||\cdot ||_{p}\) is a norm on \(L^{p}(S, \Sigma , m)\) if \(p\in [1,\infty )\), but it is not a norm for \(p<1\). This is the reason why, in Section 7.2, we will only define \(\Lp \) convergence for \(p\geq 1\).

When \(p=2\) we obtain a Hilbert space with inner product:

\[ \langle f, g \rangle = \int _{S}fg\,dm.\]

There is also a Banach space \(L^{\infty }(S, \Sigma , m)\) where

\[ ||f||_{\infty } = \inf \{M \geq 0; |f(x)| \leq M~\mbox {a.e.}\}.\]

Variants of all of these spaces exist with \(\C \) in place of \(\R \), using Lebesgue integration over \(\C \) as defined in Section 4.8. These spaces play important roles in functional analysis.