Part 6: Random Variables and Probability Section 24

The Lebesgue integral¶

The Lebesgue integral is in at least two important ways a generalization of the Riemann integral and it serves a crucial purpose of defining what it means to take the expected value of a function with respect to a probability distribution. The first sense in which the Riemann integral is generalized is that the Lebesgue integral is defined over measurable functions, a space that is much richer than that of Riemann integrable functions, the second sense is much more crucial: the Lebesgue integral is defined for functions with domain in arbitrary sets, thus allowing to handle a more abstract and general class of functions.

Intuitively the Lebesgue integral is constructed in a similar way than the Riemann integral. To construct the latter one takes successively finer grids of the domain and evaluate the function at certain points, constructing step functions, one above the function and one below, then two sums are constructed and the value of the integral is defined as the (common) value of the limit of those sums as the length of the grid's spaces goes to zero.

The Lebesgue integral of a function \(f:S\to\mathbb{R}_{+}\) is constructed by taking grids over the range of the function \(\left\lbrace y_{i}\right\rbrace _{i=1}^{n}\) such that \(0=y_{1}\leq\ldots\leq y_{n}\). Then one can define the sets \(A_{i}=\left\lbrace s\in S|y_{i}\leq f\left(s\right)<y_{i+1}\right\rbrace\) and using the measure over \(S\) define \(\lambda\left(A_{i}\right)\) and the sum \(\sum y_{i}\lambda\left(A_{i}\right)\). The Lebesgue integral is then the limit of this sum as the values \(y_{i}\) are closer together.

The introduction before of simple functions makes sense when defining the Lebesgue integral. Its definition seems intuitive for this class of functions and Proposition 23.5 creates a bridge between them and the more general class of measurable functions, thus allowing to extend the Lebesgue integral to this broader family.

In what follows we restrict attention to non-negative, real valued functions.

Definition 24.1: Lebesgue integral for simple functions

Let \(\left(S,\mathcal{A},\mu\right)\) be a measure space and \(f:S\to\mathbb{R}_{+}\) a simple, \(\mu\)-measurable function that takes no more than countably many values \(\left\lbrace y_{1},y_{2},\ldots\right\rbrace\). The Lebesgue integral over the set \(A\subseteq S\) is defined as:

\[\begin{equation} \int_{A}f\left(s\right)d\mu=\sum_{n}y_{n}\mu\left(A_{n}\right)\tag{24.1} \end{equation}\]

where the sets \(A_{n}\) are defined as:

\[ A_{n}=\left\lbrace s\in A|f\left(s\right)=y_{n}\right\rbrace \]

These sets can be empty if there is no element of \(s\) in \(A\) for which \(f\) takes a given value. The Lebesgue integral is defined as long as the series in (24.1) is absolutely convergent. If \(f\) takes finitely many values and \(\mu\) is finite (or a probability measure) this condition is satisfied.

An example is given by the constant function, \(f\left(s\right)=1\) for all \(s\in S\), then:

\[ \int_{A}f\left(s\right)d\mu=\int_{A}d\mu=\mu\left(A\right) \]

It can be shown that the lebesgue integral satisfies some natural properties:

Proposition 24.1

Let \(f\) and \(g\) be non-negative, measurable, simple and integrable functions on \(\left(S,\mathcal{A},\mu\right)\), a measure space, and \(c\geq0\) a constant. Then:

\(\int_{A}\left(f+g\right)\left(s\right)d\mu=\int_{A}f\left(s\right)d\mu+\int_{A}g\left(s\right)d\mu\)
\(\int_{A}\left(cf\right)\left(s\right)d\mu=c\int_{A}f\left(s\right)d\mu\)
If \(f\) is bounded \(\left|f\left(s\right)\right|\leq M\) a.e. then \(f\) is integrable and \(\left|\int_{A}f\left(s\right)d\mu\right|\leq M\mu\left(A\right)\).

Proof

Kolmogorov and Fomin (2012, Sec. 29.1).

Definition 24.2: Lebesgue integral - Nonnegative functions

Let \(\left(S,\mathcal{A},\mu\right)\) be a measure space. A measurable function \(f:S\to\mathbb{R}\) is said to be integrable on a set \(A\) if there exists a sequence \(\left\lbrace f_{n}\right\rbrace\) of integrable simple functions converging uniformly to \(f\) on \(A\). The Lebesgue integral is defined as:

\[\begin{equation} \int_{A}f\left(s\right)d\mu=\lim\int_{A}f_{n}\left(s\right)d\mu \tag{24.2} \end{equation}\]

This definition precludes the integral from being infinite, as shown in Kolmogorov and Fomin (2012, Sec. 29.1), the limit above exists provided that the functions \(f_{n}\) are integrable (recall that it was asked of the sum in (24.1) to be finite), moreover it is independent of the choice of sequence approximating \(f\), this sequence can be furthermore be chosen to be strictly increasing (Stokey et al. 1989). Yet, the concept of the Lebesgue integral can be easily generalized to allow for infinite values, the definition in Stokey et al. (1989) allows for this.

What follows is a list of properties of the Lebesgue integral which should be familiar if there is any knowledge of the behavior of Riemann integrals. They are not of particular interest in this course.

Proposition 24.2

Properties of the Lebesgue integral for non-negative measurable functions:

\(\int_{A}\left(f+g\right)\left(s\right)d\mu=\int_{A}f\left(s\right)d\mu+\int_{A}g\left(s\right)d\mu\)
\(\int_{A}\left(cf\right)\left(s\right)d\mu=c\int_{A}f\left(s\right)d\mu\)
If \(g\) is measurable and integrable and \(f\) is bounded by \(g\): \(\left|f\left(s\right)\right|\leq g\left(s\right)\) a.e., then \(f\) is integrable and \(\left|\int_{A}f\left(s\right)d\mu\right|\leq\int_{A}g\left(s\right)d\mu\).
1. If \(f\) is bounded and measurable then it is integrable.
If \(f\leq g\) a.e. then \(\int f\left(s\right)d\mu\leq\int g\left(s\right)d\mu\).
If \(A\subseteq B\) with \(A,B\in\mathcal{A}\) then \(\int_{A}f\left(s\right)d\mu\leq\int_{B}f\left(s\right)d\mu\)
Let \(A=\cup A_{n}\) where \(\left\lbrace A_{n}\right\rbrace\) is a finite or countable sequence of disjoint sets. If \(f\) is integrable on \(A\) then \(f\) is integrable on \(A_{n}\) for all \(n\) and:

\[ \int_{A}f\left(s\right)d\mu=\sum_{n}\int_{A_{n}}f\left(s\right)d\mu \]

when the series on the right is absolutely convergent.

Finally it is noted that a non-negative integrable function induces a measure on a space, the following proposition makes this clear.

Proposition 24.3

Let \(f\) be a non-negative, integrable function, then \(\lambda:\mathcal{A}\to\mathbb{R}\) defined as:

\[ \lambda\left(A\right)=\int_{A}f\left(s\right)d\mu \]

is a measure on \(\left(S,\mathcal{A}\right)\).

Definition 24.3: Lebesgue integral

Let \(\left(S,\mathcal{A},\mu\right)\) be a measure space. A measurable function \(f:S\to\mathbb{R}\) is said to be integrable if the following two integrals are finite:

\[ \int f^{+}\left(s\right)d\mu\qquad\int f^{-}\left(s\right)d\mu \]

where:

\[ f^{+}\left(s\right)=\begin{cases} f\left(s\right) & if f\left(s\right)\geq0\\ 0 & if f\left(s\right)<0 \end{cases}\qquad f^{+}\left(s\right)=\begin{cases} 0 & if f\left(s\right)\geq0\\ -f\left(s\right) & if f\left(s\right)<0 \end{cases} \]

The integral of \(f\) is defined as:

\[\begin{equation} \int f\left(s\right)d\mu=\int f^{+}\left(s\right)d\mu-\int f^{-}\left(s\right)d\mu \tag{24.3} \end{equation}\]

Recall that when \(\left(S,\mathcal{A},\mu\right)\) is a probability space the function \(f\) is called a random variable, the definitions above are then the definitions of the expected value of a random variable, this expected value exists when \(f\) is integrable, we have seen that a sufficient condition for this is to be bounded a.e. and the measure to be finite, this last condition is satisfied immediately by probability measures.