Abstract Integration
Lebesgue integration is sometimes referred to as abstract integration. The reason is that the
Lebesgue integral is developed in the very general setting of a measure space \((\Omega, \mathcal{F}, \mu)\).
We want to define the integral:
\[
\int_{\Omega} g(\omega) d\mu(\omega)
\]
of a measurable function \(g: \Omega \to \overline{\mathbb{R}}\) defined on a measure space \((\Omega, \mathcal{F}, \mu)\).
Note: \(\overline{\mathbb{R}}\) refers to the extended set of real values, which includes \(\infty\) and \(-\infty\).
Let's check some special cases:
If the measure space is a probability space:\((\Omega, \mathcal{F}, \mathbb{P})\), and \(X: \Omega \to \bar{\mathbb{R}}\) is
measurable, which means \(X\) is an extended-valued random variable, then the integral is the expectation of \(X\):
\[
\int_{\Omega} X \, d\mathbb{P} = \mathbb{E }(X).
\]
If we define the measure space as \((\mathbb{R}, \mathcal{B}, \lambda)\) where \(\mathcal{B}\) is the Borel
\(\sigma\)-algebra and \(\lambda\) is he Lebesgue measure, then the integral is a generalization of the usual integral
encountered in calculus:
\[
\int_{\mathbb{R}} g \, d\lambda = \int g(x) dx.
\]
So, even if it isn't always stated explicitly, many of the integrals we have encountered — whether in the context of computing
expectations in probability theory or evaluating integrals on the real line — can be defined using Lebesgue integration.
This approach is the core of modern analysis and probability, providing a robust framework for handling functions that may
be too irregular or complex for the Riemann integration.
Characteristic function
The integral over a measurable subset \(B\) of \(g\) is defined by
\[
\int_B g \, d\mu = \int (\chi_B \, g)d\mu
\]
where \(\chi_B\)(or alternatively, \( 1_B\)) is a characteristic function(or indicator function) so that
\[
(\chi_B \, g) (\omega) = \begin{cases}
g(\omega) & \text{if \(\omega \in B\)}\\
0 & \text{if \(\omega \notin B\)}.
\end{cases}
\]
We will use the term "almost everywhere"(a.e.) to mean "for all \(\omega\) outside
a zero-measure subset of \(\Omega\). So, we "ignore" a set of \(\omega\)'s that has measure zero.
For the special case of probability measure, we also use "almost surely"(a.s.).
For example,
\[
g_n \uparrow g, \, a.e.
\]
means that the increasing monotonic convergence of \(g_n(\omega)\) to \(g(\omega)\) holds \(\forall \omega \in \Omega\)
outside a zero-measure set.
The Integral of Finite Nonnegative Measurable Functions
A function \(g: \Omega \to \mathbb{R}\) is said to be simple if it
is measurable, finite and takes only finitely many different values.
If \(g\) is a simple function of the form:
\[
g(\omega) = \sum_{i=1}^k a_i \chi_{A_i}(\omega), \quad \forall \omega \in \Omega
\]
where \(k\) is a finite nonnegative integer, \(a_i \in \mathbb{R}\) and \(A_i\) are measurable sets.
Then its integral is defined by
\[
\int g \, d\mu = \sum_{i=1}^k a_i \mu(A_i).
\]
(We assume \(a_i \mu(A_i) = 0\) if \(a_i = 0\) and \(\mu(A_i) = \infty\).)
Example: Dirichlet function
Finally, we can get the integral of Dirichlet function on the interval \([0, 1]\):
\[
f(x)=
\begin{cases}
1 &\text{if \(x \in \mathbb{Q}\)} \\
0 &\text{if \(x \in \mathbb{R} \setminus \mathbb{Q}\)}
\end{cases}
\]
The Dirichlet function is a simple function since it only takes on the two values 0 and 1. By the definiton:
\[
\begin{align*}
\int_0^1 f(x)dx &= \int_0^1 \chi_{\mathbb{Q}}(x) dx\\\\
&= 1 \cdot \mu([0, 1] \cap \mathbb{Q})(x) + 0 \cdot \mu([0, 1] \setminus \mathbb{Q})(x)\\\\
&= 1 \cdot 0 + 0 \cdot (1 - 0) \\\\
&= 0
\end{align*}
\]
Note: Technically, for any interval \([a, b]\), the Lebesgue integral of the Dirichlet function is 0
because the Lebesgue measure of any countable set is zero.
The Integral of Nonnegative Measurable Functions
We approximate the integral of a nonnegative function \(g\) using simple functions.
For a nonnegative (extended-valued) measurable function \(g: \Omega \to [0, \infty]\), we let \(S(g)\) be the
set of all nonnegative simple (hence automatically measurable) functions \(q\) that satisfy \(0 \leq q \leq g\), and
define
\[
\int g \, d\mu = \sup_{q \in S(g)} \int q \, d\mu.
\]
The Integral of General Measurable Functions
Consider a measurable function \(g: \Omega \to \overline{\mathbb{R}}\). Let
\[
A_+ = \{\omega | g(\omega) > 0\}, \qquad g_+ = g \cdot \chi_{A_+}
\]
and
\[
A_- = \{\omega | g(\omega) < 0\}, \qquad g_- = g \cdot \chi_{A_-}
\]
Note that \(A_+\) , \(A_-\) are measurable sets, and \(g_+\) , \(g_-\) are nonnegative(possibly extended-valued)
measurable functions.
Then we have \(g = g_+ - g_-\) and define
\[
\int g \, d\mu = \int g_+ \, d\mu - \int g_- \, d\mu
\]
if we have both \(\int g_+ \, d\mu < \infty\) and \(\int g_- \, d\mu < \infty\).
Note: The definition implies there exists a function that is NOT Lebesgue integrable over some interval. In general,
if a function is Riemann integrable, the function is also Lebesgue integrable and
\[
\int_{[a, b]} f(x) dx = \int_a^b f(x) dx
\]
but some improperly Riemann integrable function cannot be Lebesgue integrable.
Example: Sinc function over \([0, \infty)\)
Consider the Dirichlet integral:
\[
\int_0^{\infty} \frac{\sin x}{x} dx
\]
This is known to converge to \(\frac{\pi}{2}\):
\[
\begin{align*}
\int_0^{\infty} \frac{\sin x}{x} dx &= \lim_{b \to \infty} \int_0^b \frac{\sin x}{x} dx \\\\
&= \frac{\pi}{2}
\end{align*}
\]
So, \(f\) is improperly Riemann integrable on \([0, \infty)\).
On the other hand, in the Lebesgue sense:
\[
\int_0^{\infty} \frac{\sin x}{x} dx = \int_0^{\infty} \left(\frac{\sin x}{x}\right)_+ dx - \int_0^{\infty} \left(\frac{\sin x}{x}\right)_- dx
\]
\(\sin x\) is always positive on the interval \([2\pi n, 2\pi n + \pi]\), and then
\[
\begin{align*}
\int_0^{\infty} \left(\frac{\sin x}{x}\right)_+ dx &= \sum_{n=0}^{\infty} \int_{2\pi n}^{2\pi n + \pi} \frac{\sin x}{x} dx \\\\
&\geq \sum_{n=0}^{\infty} \int_{2\pi n}^{2\pi n + \pi} \frac{\sin x}{2 \pi n + \pi} dx \\\\
&= \sum_{n=0}^{\infty} \frac{2}{\pi(2n +1)} \\\\
&= \infty
\end{align*}
\]
and similarly, \(\int_0^{\infty} \left(\frac{\sin x}{x}\right)_- dx = \infty\).
Thus, by the definition, this is NOT integrable in the Lebesgue sense.