Probability with Measure

4.4 Integration as a measure

In Lemma 4.1.4 we saw that integrals of simple functions gave us a way of constructing measures. We’ll now carry that property over to integrals of non-negative functions. In this case the upgrade to non-negative measurable functions provides the final version of the property. Other properties, such as monotonicity and linearity, will receive one more upgrade in Section 4.5. We continue to work over a general measure space \((S,\Sigma ,m)\).

Theorem 4.4.1 Let \(f:S\to [0,\infty )\) be measurable. Then \(\nu :\Sigma \to [0,\infty ]\) by

\[\nu (A)=\int _A f\,dm\]

is a measure.

Proof: We check the two properties in Definition 3.1.1. We have \(\nu (\emptyset )=\int _\emptyset f\,dm =\int _S \1_{\emptyset } f\,dm\). Since \(\1_{\emptyset }=0\) this gives \(\nu (\emptyset )=\int _S 0\,dm\). The zero function is a simple function \(0=0\1_S\), and (4.3) gives that it has integral zero. Thus \(\nu (\emptyset )=0\).

We need to show that \(\nu \) is countably additive. Let \((E_n)_{n\in \N }\) be pairwise disjoint subsets of \(S\) and let \(E=\bigcup _{n=1}^\infty E_n\). Set \(F_n=\bigcup _{i=1}^n E_i\). Then \(F_n\sw F_{n+1}\) and hence \(\1_{F_n}\leq \1_{F_{n+1}}\), so \(\1_{F_n}f\leq \1_{F_{n+1}}f\). Also, \(\bigcup _{n=1}^\infty F_n=\bigcup _{n=1}^\infty E_n\), so \(\1_{F_n}\to \1_{E}\) pointwise. Hence \(\1_{F_n}f\to \1_{E}f\) pointwise so by Theorem 4.3.1 we have

\begin{equation} \label {eq:int_as_meas_M2_pre} \int _S \1_{F_n}f\,dm \to \int _S \1_{E}f\,dm. \end{equation}

The right hand side of the above is equal to \(\int _E f\,dm=\nu (E)\). By Lemma 4.3.2 the left hand side is equal to

\[\int _S\sum _{i=1}^n \1_{E_i}f\,dm = \sum _{i=1}^n\int _S\1_{E_i}f\,dm=\sum _{i=1}^n\int _{E_i} f\,dm = \sum _{i=1}^n\nu (E_i).\]

Putting these into (4.14) gives that \(\lim _n \sum _{i=1}^n \nu (E_i) = \nu (\bigcup _{i=1}^\infty E_i)\), as required. ∎

Example 4.4.2 The Gaussian measure on \(\R \) is obtained by taking \(f=\phi \) where \(\phi :\R \to \R \) is given by \(\phi (x) = \frac {1}{\sqrt {2\pi }}e^{-x^2/2}\) and taking \(m\) as Lebesgue measure. We note an explicit connection with probability theory:

\[I_A(\phi )=\int _A \frac {1}{2\pi }e^{-x^2/2}\,d\lambda (x)\]

which you should recognize as equal to \(\P [Z\in A]\) where \(Z\sim N(0,1)\). Thus \(A\mapsto \int _A \phi \,d\lambda \) is the law of a standard normal random variable. Normal random variables are often known as Gaussian random variables.