Introduction
When we developed Riemann integrability,
we observed that certain functions fail to be integrable in either the Riemann or the improper Riemann sense.
To handle such cases, we need to introduce measure theory and Lebesgue
integration. Since our focus is on applied mathematics, particularly in the context of statistics and
machine learning, we will concentrate on probability-related measure theory. While we will avoid covering
every foundational topic and formal mathematical detail, our goal is to build a solid understanding that leads
to the introduction of Lebesgue integration.
Note that measure theory fundamentally defines the "volume" and structure of a space.
This perspective is crucial when transitioning from discrete symmetries to continuous ones, such as \(SE(3)\),
where the measure remains invariant under rotation.
(See: geometry of symmetry)
Probability Space
While the intuitive notion of "volume" or "probability" works well for simple shapes and finite sets, it
becomes surprisingly fragile once we pass to the continuum of real numbers. To avoid logical paradoxes there,
we must move beyond mere intuition and adopt a rigorous axiomatic framework.
In this context, we do not just "measure" things. We define a structured environment in which every operation
is logically consistent. This is why we treat a probabilistic model not as a single value, but as a triple of
interconnected components.
Now, we need a formal definition of probabilistic model:
Definition: Probability Space
A probability space is a triple \((\Omega, \mathcal{F}, \mathbb{P})\) where
- \(\Omega\) is the sample space: the set of possible outcomes of an
experiment.
- \(\mathcal{F}\) is a \(\sigma\)-algebra: a collection of subsets of the
sample space \(\Omega\).
- \(\mathbb{P}\) is a probability measure: a function
\(\mathbb{P}: \mathcal{F} \to [0, 1]\) satisfying \(\mathbb{P}(\Omega) = 1\) together
with countable additivity (defined formally below).
Sample Space
The sample space \(\Omega\) can be finite, countable, or uncountable. An element of \(\Omega\) is denoted by
\(\omega\), and is called an elementary outcome.
For example, if our experiment consists of an infinite number of consecutive rolls of a die, the sample space
is the set:
\[
\Omega = \{1, 2, 3, 4, 5, 6\}^{\infty}
\]
and an elementary outcome is an infinite sequence such as:
\[
\omega = (1, 1, 4, 3, 1, 5, \ldots ).
\]
A simpler case of the probability space can be a discrete probability space. In this case,
the sample space is finite, or countable. When it is countable we enumerate the outcomes as
\[
\Omega = \{\omega_1, \omega_2, \ldots \}
\]
and a finite sample space is the same picture with the list stopping at some \(\omega_N\). Either way, the
\(\sigma\)-algebra is the set of all subsets of \(\Omega\). Then the probability measure assigns a
number in the set \([0, 1]\) to every subset of \(\Omega\). It is defined in terms of the probabilities
\(\mathbb{P}(\{\omega\})\) of the elementary outcomes and satisfies
\[
\forall A \subset \Omega, \quad \mathbb{P}(A) = \sum_{\omega \in A}\mathbb{P}(\{\omega\})
\]
and
\[
\sum_{\omega \in \Omega}\mathbb{P}(\{\omega\}) = 1.
\]
We write \(\mathbb{P}(\omega)\) for \(\mathbb{P}(\{\omega\})\), and we abbreviate \(\mathbb{P}(\omega_i)\) as
\(p_i\).
\(\sigma\)-algebra (\(\sigma\)-field)
Ideally, we wish to specify the probability \(\mathbb{P} (A)\) of "every" subset of \(\Omega\). However, this
is too complicated, especially when \(\Omega\) is uncountable. So, we assign probabilities to only a partial
collection of subsets of \(\Omega\). The sets in this collection are to be thought of as the "nice" and
"interesting" subsets of \(\Omega\). Formally, we define the collection as follows:
Definition: \(\sigma\)-algebra
A \(\sigma\)-algebra \(\mathcal{F}\) is a collection of subsets of \(\Omega\) with the
following properties:
- \(\emptyset \in \mathcal{F}\).
- If \(A \in \mathcal{F}\), then \(A^c \in \mathcal{F}\).
- If \(\{A_i\}_{i=1}^{\infty} \subset \mathcal{F}\), then \(\bigcup_{i =1}^{\infty} A_i \in \mathcal{F}\).
An event \(A\) is called an \(\mathcal{F}\)-measurable set, or simply a measurable set.
The pair \((\Omega, \mathcal{F})\) is called a measurable space.
Given a collection \(\mathcal{C}\) of subsets of \(\Omega\), we define the \(\sigma\)-algebra \(\mathcal{F}\)
as the intersection of all \(\sigma\)-algebras that contain \(\mathcal{C}\), since what we need is the
smallest \(\sigma\)-algebra containing \(\mathcal{C}\). In this manner, \(\mathcal{F}\) is
said to be the \(\sigma\)-algebra generated by \(\mathcal{C}\), and is denoted by \(\sigma(\mathcal{C})\).
Probability Measure
A collection of sets \(A_{\alpha} \subset \Omega\), where \(\alpha\) ranges over some index set, is
mutually exclusive, or disjoint, if
\(A_{\alpha} \cap A_{\alpha'} = \emptyset\) whenever \(\alpha \neq \alpha'\). Also, the sets
\(A_{\alpha} \subset \Omega\) are called collectively exhaustive if
\(\bigcup_{\alpha} A_{\alpha} = \Omega\).
Definition: Measure
A measure is a function
\[
\mu: \mathcal{F} \to [0, \infty]
\]
which assigns a nonnegative extended real number \(\mu(A)\) to every set \(A \in \mathcal{F}\), and which satisfies the
following two conditions:
- \(\mu(\emptyset) = 0\).
- Countable additivity (\(\sigma\)-additivity): if \(\{A_i\}\) is a sequence
of disjoint sets that belong to \(\mathcal{F}\), then
\[
\mu(\bigcup_i A_i) = \sum_{i =1}^{\infty} \mu(A_i).
\]
This general definition covers measures that may be infinite on the whole space. A measure with
\(\mu(\Omega) \lt \infty\) is called a finite measure. The special case \(\mu(\Omega) = 1\)
is a probability measure, defined next.
Definition: Probability Measure
A probability measure is a measure \(\mathbb{P}\) with the additional property \(\mathbb{P}(\Omega) = 1\).
A crucial consequence of countable additivity is that the probability measure is continuous.
This continuity property of measure ensures that probabilities behave well under limits,
which is essential for defining concepts like convergence of random variables and the law of large numbers.
Theorem: Continuity of Measure
Let \((\Omega, \mathcal{F}, \mathbb{P})\) be a probability space and \(\{A_n\} \subset \mathcal{F}\).
- Continuity from below. If \(A_n \uparrow A\) (that is, \(A_1 \subset A_2 \subset \cdots\) and \(\bigcup_n A_n = A\)), then
\[
\lim_{n \to \infty} \mathbb{P}(A_n) = \mathbb{P}(A).
\]
- Continuity from above. If \(A_n \downarrow A\) (that is, \(A_1 \supset A_2 \supset \cdots\) and \(\bigcap_n A_n = A\)), then
\[
\lim_{n \to \infty} \mathbb{P}(A_n) = \mathbb{P}(A).
\]
For a general measure \(\mu\), continuity from below holds unconditionally, but continuity from above
requires the additional hypothesis \(\mu(A_1) \lt \infty\) (one can construct counterexamples on
\(\mathbb{R}\) with Lebesgue measure otherwise). Probability measures automatically satisfy this since
\(\mathbb{P}(\Omega) = 1\).
Proof (sketch):
From below. Set \(B_1 = A_1\) and \(B_n = A_n \setminus A_{n-1}\) for \(n \geq 2\). The
\(B_n\) are pairwise disjoint, \(\bigcup_{k=1}^n B_k = A_n\), and \(\bigcup_{k=1}^\infty B_k = A\).
Countable additivity gives
\[
\begin{align*}
\mathbb{P}(A)
&= \sum_{k=1}^\infty \mathbb{P}(B_k) \\\\
&= \lim_{n \to \infty} \sum_{k=1}^n \mathbb{P}(B_k) \\\\
&= \lim_{n \to \infty} \mathbb{P}(A_n).
\end{align*}
\]
From above. Since \(A_n \downarrow A\), by De Morgan \(A_n^c \uparrow A^c\). Applying
continuity from below to the complements yields \(\lim_n \mathbb{P}(A_n^c) = \mathbb{P}(A^c)\). Because
\(\mathbb{P}\) is a finite measure with \(\mathbb{P}(\Omega) = 1\), we have
\(\mathbb{P}(A_n^c) = 1 - \mathbb{P}(A_n)\) and \(\mathbb{P}(A^c) = 1 - \mathbb{P}(A)\). Substituting
these gives \(\lim_n (1 - \mathbb{P}(A_n)) = 1 - \mathbb{P}(A)\), hence
\(\lim_n \mathbb{P}(A_n) = \mathbb{P}(A)\).
General measures. The argument from below used only the two defining properties of a measure,
so it applies verbatim to any \(\mu\). For the argument from above, replace the passage to complements by
\(D_n = A_1 \setminus A_n\), which increases to \(A_1 \setminus A\). Continuity from below gives
\(\lim_n \mu(D_n) = \mu(A_1 \setminus A)\). Both \(A_n\) and \(A\) sit inside \(A_1\), so once
\(\mu(A_1) \lt \infty\) every measure appearing here is finite and additivity rewrites the limit as
\(\lim_n (\mu(A_1) - \mu(A_n)) = \mu(A_1) - \mu(A)\). Cancelling \(\mu(A_1)\) leaves
\(\lim_n \mu(A_n) = \mu(A)\), and that cancellation is the step that fails when \(\mu(A_1) = \infty\).
Countable additivity implies that probabilities (and more generally, measures) behave like the notion of
volume. The volume of a countable union of disjoint sets is the sum of their individual "volumes." Indeed, a
measure is a generalized notion of a volume that characterizes the "size" of sets within a
manifold.
In physical systems like robotics \(SE(3)\), this ensures that the core properties of an object are preserved
throughout any rigid body motion.
\(\sigma\)-Finite Measures
The continuity-from-above argument above required the finiteness hypothesis \(\mu(A_1) \lt \infty\), which is
automatic for probability measures but fails for general measures such as Lebesgue measure on \(\mathbb{R}\).
For many results in measure theory and its applications, we do not need full finiteness. It suffices that the
space can be exhausted by countably many finite-measure pieces.
Definition: \(\sigma\)-Finite Measure
A measure \(\mu\) on \((\Omega, \mathcal{F})\) is \(\sigma\)-finite if there exists a
countable collection \(\{\Omega_n\}_{n \geq 1} \subseteq \mathcal{F}\) (not required to be disjoint) with
\(\Omega = \bigcup_{n=1}^\infty \Omega_n\) and \(\mu(\Omega_n) \lt \infty\) for every \(n\).
Every finite measure is \(\sigma\)-finite (take \(\Omega_1 = \Omega\)), and in particular every probability
measure is \(\sigma\)-finite. The Lebesgue measure on \(\mathbb{R}\) is \(\sigma\)-finite but not finite, via
the exhaustion \(\Omega_n = [-n, n]\). By contrast, the counting measure on an uncountable set \(\Omega\) is
not \(\sigma\)-finite. Any \(\Omega_n\) with finite counting measure is a finite set, and a countable
union of finite sets is countable, which cannot cover an uncountable \(\Omega\).
The \(\sigma\)-finite hypothesis is a standard regularity condition appearing throughout measure theory. It is
the weakest assumption under which many fundamental results hold, and we will invoke it in later pages
whenever our setting calls for reasoning that strictly exceeds the finite-measure case.
Finite Additivity
At this point, the \(\sigma\)-algebra and probability measure definitions look quite
abstract. Do we really need all of this machinery just to define probabilities?
The answer lies in the concept of countable additivity (\(\sigma\)-additivity), which requires
handling infinite sequences of disjoint events. This property is essential for dealing with limits
and continuous spaces (like the real line). The difficulty is not the property itself but the collection we must
impose it on, since a \(\sigma\)-algebra is specified only indirectly, as the smallest one containing something
simpler.
Fortunately, that indirect specification can be avoided at the outset. There is a structure closed under
finite unions whose members we can write down explicitly, and it serves as a stepping stone. We
define probabilities on it and then let Carathéodory's Extension Theorem carry them to the full
\(\sigma\)-algebra.
This approach is analogous to defining a function on a dense subset and then extending it by continuity. We start
with the sets we can describe by hand, and the extension theorem supplies the rest.
Definition: Algebra (Field)
An algebra (or, a field) is a collection \(\mathcal{F}_0\) of subsets of \(\Omega\) with the following properties:
- \(\emptyset \in \mathcal{F}_0\).
- If \(A \in \mathcal{F}_0\), then \(A^c \in \mathcal{F}_0\).
- If \(A, B \in \mathcal{F}_0\), then \(A \cup B \in \mathcal{F}_0\).
Notice that this is identical to the \(\sigma\)-algebra definition, except property (3) only requires closure
under finite unions rather than countable unions. This makes it far easier to verify in practice.
Definition: Finite Additivity
A function \(\mathbb{P}: \mathcal{F}_0 \to [0, 1]\) is said to be finitely additive if
\[
\begin{align*}
&A, B \in \mathcal{F}_0 , \quad A \cap B = \emptyset \\\\
&\Longrightarrow \mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B).
\end{align*}
\]
Every \(\sigma\)-additive function is automatically finitely additive, but finite additivity alone is not
strong enough to force countable additivity. Carathéodory's Extension Theorem (below) does not close
that gap, since it takes \(\sigma\)-additivity on the algebra as a hypothesis. What the theorem removes is a
different difficulty. A \(\sigma\)-additive function defined on a small algebra can be uniquely extended to a
full measure on the generated \(\sigma\)-algebra.
We examine Carathéodory's Extension Theorem next.
Carathéodory's Extension Theorem
Defining a probability measure directly on a full \(\sigma\)-algebra is often technically overwhelming because
\(\sigma\)-algebras contain incredibly complex sets (formed by countable unions and limits). However, it is
usually straightforward to define probabilities on a smaller, simpler structure called an
algebra (for example, finite unions of intervals).
The following theorem acts as a powerful bridge. It guarantees that if we can essentially
"get the definitions right" on the simple building blocks (the algebra), the mathematics automatically and
uniquely extends that definition to the entire complex \(\sigma\)-algebra.
Theorem: Carathéodory's Extension Theorem
Let \(\mathcal{F}_0\) be an algebra of subsets of a sample space \(\Omega\), and let
\(\mathcal{F} = \sigma(\mathcal{F}_0)\) be the \(\sigma\)-algebra that it generates.
Suppose that \(\mathbb{P}_0 : \mathcal{F}_0 \to [0, 1]\) satisfies \(\mathbb{P}_0(\Omega) = 1\) and is
\(\sigma\)-additive on \(\mathcal{F}_0\). Concretely, whenever
\(\{A_i\}_{i=1}^{\infty} \subset \mathcal{F}_0\) are pairwise disjoint with
\(\bigcup_{i=1}^{\infty} A_i \in \mathcal{F}_0\), we have
\(\mathbb{P}_0\left(\bigcup_{i=1}^{\infty} A_i\right) = \sum_{i=1}^{\infty} \mathbb{P}_0(A_i)\).
Then, \(\mathbb{P}_0\) can be extended uniquely to a probability measure on \((\Omega, \mathcal{F})\).
That is, there exists a unique probability measure \(\mathbb{P}\) on \((\Omega, \mathcal{F})\) such that
\[
\forall A \in \mathcal{F}_0, \quad \mathbb{P}(A) = \mathbb{P}_0(A).
\]
The proof is substantial and belongs to the real-analysis track, so we defer it until that later
development. It constructs an outer measure from \(\mathbb{P}_0\), identifies the family of
Carathéodory-measurable sets, and establishes uniqueness by a monotone-class argument.
This theorem provides the fundamental existence and uniqueness guarantee for measures.
Without it, we would have to manually prove the existence of a measure for every complicated Borel set, which
is practically impossible. It assures us that our intuitive notion of "length" on intervals can be rigorously
extended to a full-fledged measure without logical contradictions. In the next section, we will directly apply
this theorem to construct the Lebesgue measure from simple interval lengths.
Lebesgue measure
The uniform distribution on the interval \([0, 1]\) assigns probability \(b - a\) to every
interval \([a, b] \subset [0, 1]\). We want to define the appropriate \(\sigma\)-algebra and the probability
measure on the sample space \(\Omega = [0, 1]\), but first, we consider the sample space:
\[
\Omega' = (0, 1].
\]
Definition: Borel \(\sigma\)-algebra
Consider the collection \(\mathcal{C}\) of all intervals \((a, b]\) contained in \((0, 1]\) and let \(\mathcal{F}\) be
the \(\sigma\)-algebra generated by \(\mathcal{C}\). This is called the Borel \(\sigma\)-algebra and is
denoted by \(\mathcal{B}\). Every set that belongs to this \(\sigma\)-algebra is called a Borel (measurable) set.
Any set that can be formed by starting with intervals \((a, b]\) is a Borel set. For example, the set of
rational numbers in \((0, 1]\), and its complement, the set of irrational numbers in \((0, 1]\), are Borel
sets.
Since defining a probability measure for all Borel sets is too complicated, we start with a "smaller"
collection, \(\mathcal{F}_0 \subset \mathcal{B}\). We let \(\mathcal{F}_0\) consist of the empty set and all sets
that are finite unions of disjoint intervals of the form \((a, b]\). For example,
\[
\begin{align*}
&A \in \mathcal{F}_0 \\\\
&\Longrightarrow A = (a_1, b_1] \cup (a_2, b_2] \cup \cdots \cup (a_n, b_n],
\end{align*}
\]
where \(0 \leq a_1 \lt b_1 \leq a_2 \lt b_2 \leq \cdots \leq a_n \lt b_n \leq 1, \quad n \in \mathbb{N}\).
This collection is an algebra. It contains the empty set by fiat, the complement in \((0, 1]\) of a finite
disjoint union of intervals \((a, b]\) is again such a union, and the union of two members can be rewritten as
a disjoint union by merging the intervals that overlap. Moreover
\(\mathcal{C} \subset \mathcal{F}_0 \subset \sigma(\mathcal{C})\), so
\(\sigma(\mathcal{F}_0) = \sigma(\mathcal{C}) = \mathcal{B}\) and the algebra generates exactly the Borel
\(\sigma\)-algebra.
Also, we define:
\[
\mathbb{P}_0(A) = (b_1 - a_1) + (b_2 - a_2) + \cdots + (b_n - a_n).
\]
Splitting or merging at a shared endpoint gives more than one admissible representation of the same set \(A\),
but the value is unaffected, since replacing \((a, b]\) by \((a, c] \cup (c, b]\) replaces \(b - a\) by
\((c - a) + (b - c)\). Finite additivity of \(\mathbb{P}_0\) is immediate from the same formula, since
adjoining a disjoint interval adds its length to the sum. Countable additivity on \(\mathcal{F}_0\) is a
different matter. Its essential case asserts that whenever a single interval \((a, b]\) is split into
countably many disjoint intervals \((a_i, b_i]\), the lengths still add up to \(b - a\), and establishing that
requires a compactness argument for closed bounded intervals. We take the property on faith here, as the one
substantive hypothesis behind the construction, and leave its verification alongside the deferred proof of the
extension theorem.
Granting that hypothesis, the requirements of Carathéodory's Extension Theorem are met, since
\(\mathbb{P}_0\) also gives the whole sample space total mass \(\mathbb{P}_0((0, 1]) = 1\). The theorem
produces a probability measure \(\mathbb{P}\) defined on the entire Borel \(\sigma\)-algebra \(\mathcal{B}\)
that agrees with \(\mathbb{P}_0\) on \(\mathcal{F}_0\). We call this measure the Lebesgue or
uniform measure. In particular,
\[
\forall \, (a, b] \subset (0, 1], \quad \mathbb{P}((a, b]) = b - a.
\]
Definition: Lebesgue Measure
The probability measure \(\mathbb{P}\) on \(((0, 1], \mathcal{B})\) constructed above via
Carathéodory's Extension Theorem is called the Lebesgue measure (also
uniform measure) on \((0, 1]\). It is the unique measure satisfying
\[
\mathbb{P}((a, b]) = b - a \quad \text{for all } (a, b] \subset (0, 1].
\]
The construction extends to \(\mathbb{R}\) by the countable-additivity gluing described later in this
section. The result is a (non-probability) measure \(\mu\) defined on the Borel \(\sigma\)-algebra of
\(\mathbb{R}\), the one generated by the intervals \((a, b] \subset \mathbb{R}\), and
\(\mu((a, b]) = b - a\) holds for every such interval.
Mathematical Detail: Borel vs Lebesgue Measure
Strictly speaking, the measure constructed above on the Borel \(\sigma\)-algebra \(\mathcal{B}\) is called
the Borel measure. The Lebesgue measure, in its strictest sense, is
defined on a slightly larger \(\sigma\)-algebra called the Lebesgue \(\sigma\)-algebra,
which is the completion of \(\mathcal{B}\) with respect to the Borel measure. This
completion ensures that every subset of a set with measure zero is itself measurable (and has measure
zero).
For most practical applications in CS and statistics, this distinction is subtle. The two measures agree
on every Borel set, and integrals of measurable functions are unchanged. However, the completed version is
the natural setting for the "almost everywhere" identifications underlying \(L^p\) spaces and the
Radon-Nikodym theorem.
Finally, by adding \(\{0\}\) to the sample space \(\Omega'\), and assigning zero probability to it, we obtain
the uniform distribution model with sample space \(\Omega = [0, 1]\). (We only need to check that \(\{0\}\) is
measurable, and it is a Borel set.)
Revisit our Problem:
Recall the problem we met when the Riemann integral was introduced:
Consider the Dirichlet function:
\[
f(x)=
\begin{cases}
1 &\text{if \(x \in \mathbb{Q}\)} \\\\
0 &\text{if \(x \in \mathbb{R} \setminus \mathbb{Q}\)}
\end{cases}
\]
The integral at issue is
\[
\int_0^1 f(x)dx.
\]
Now, we can see the interval \([0, 1]\) as the sample space \(\Omega\). The smallest \(\sigma\)-algebra
that contains every interval \((a, b] \subset [0, 1]\) is the Borel \(\sigma\)-algebra.
We claim the Dirichlet function is measurable with respect to it, and check three steps:
- Every singleton \(\{q\}\) with \(q \gt 0\) is Borel, since
\(\{q\} = \bigcap_{n \geq 1/q} (q - 1/n, q]\) exhibits it as a countable intersection of generating
intervals, and a \(\sigma\)-algebra is closed under countable intersections by De Morgan. The remaining
singleton \(\{0\}\) is the complement of \((0, 1]\) in \(\Omega\).
- \(\mathbb{Q} \cap [0, 1]\) is Borel, as a countable union
\(\bigcup_{q \in \mathbb{Q} \cap [0,1]} \{q\}\) of Borel singletons.
- The Dirichlet function is the indicator of this Borel set. The indicator of a Borel set is
automatically Borel-measurable. The next page on Lebesgue integration pins down this
fact.
Hence the Dirichlet function is Borel-measurable.
We are getting closer to Lebesgue integration, but we would like to learn more about Lebesgue measure.
Construction: Lebesgue Measure on \(\mathbb{R}\)
Consider the sample space \(\Omega = \mathbb{R}\). As usual, we define a \(\sigma\)-algebra of subsets of
\(\mathbb{R}\). Let \(\mathcal{C}\) be the collection of all intervals of the form \((a, b]\) and write
\(\mathcal{B} = \sigma(\mathcal{C})\) for the \(\sigma\)-algebra it generates.
Let \(\mathbb{P}_n\) be the uniform measure on \((n, n+1]\). Given a Borel set \(A \in \mathcal{B}\), we
decompose it into countably many pieces, each piece contained in some interval \((n, n+1]\), and define
its "length" \(\mu(A)\) using countable additivity as follows:
\[
\mu(A) = \sum_{n = -\infty}^{\infty} \mathbb{P}_n \left(A \cap (n, n+1] \right).
\]
Since \(A \cap (n, n+1]\) is a measurable subset of \((n, n+1]\),
\(\mathbb{P}_n (A \cap (n, n+1]) \geq 0\). Thus, nonnegativity holds:
\[
\mu(A) \in [0, \infty].
\]
Also, \(\emptyset \cap (n, n+1] = \emptyset\) and \(\mathbb{P}_n ( \emptyset) = 0\), thus
\[
\mu(\emptyset) = 0.
\]
Now, we need to check countable additivity. Let \(\{A_i\}_{i=1}^{\infty}\) be a countable collection of
pairwise disjoint sets in \(\mathcal{B}\). For each fixed \(n\), the sets
\(\left\{A_i \cap (n, n+1] \right\}_{i=1}^{\infty}\) are also pairwise disjoint. Since \(\mathbb{P}_n\) is
the uniform measure on \((n, n+1]\), it satisfies countable additivity. That is,
\[
\begin{align*}
&\mathbb{P}_n \left(\bigcup_{i=1}^{\infty} (A_i \cap (n, n+1] )\right) = \sum_{i=1}^{\infty} \mathbb{P}_n \left(A_i \cap (n, n+1] \right) \\\\
&\Longrightarrow \mathbb{P}_n \left( \left( \bigcup_{i=1}^{\infty} A_i \right) \cap (n, n+1] \right) = \sum_{i=1}^{\infty} \mathbb{P}_n \left(A_i \cap (n, n+1]\right).
\end{align*}
\]
Then we have:
\[
\begin{align*}
\mu\left( \bigcup_{i=1}^{\infty} A_i \right) &= \sum_{n=-\infty}^{\infty} \mathbb{P}_n \left(\left( \bigcup_{i=1}^{\infty} A_i \right) \cap (n, n+1] \right) \\\\
&= \sum_{n= -\infty}^{\infty} \sum_{i = 1}^{\infty} \mathbb{P}_n (A_i \cap (n, n+1] ) \\\\
&= \sum_{i = 1}^{\infty} \sum_{n= -\infty}^{\infty} \mathbb{P}_n (A_i \cap (n, n+1] ) \\\\
&= \sum_{i = 1}^{\infty} \mu(A_i).
\end{align*}
\]
(We can exchange the order of summation because the terms in the sums are nonnegative. The interchange is
Tonelli's theorem for counting measures, and in elementary terms a direct consequence of the convergence
of nonnegative-term double series.) Therefore, \(\mu\) is a measure on \((\mathbb{R}, \mathcal{B})\),
which is the Lebesgue measure.
In discrete probability, an elementary outcome \(\omega\) has positive mass.
However, in the Lebesgue measure on \(\mathbb{R}\), every singleton set \(\{x\}\) has measure zero.
Using the continuity of measure (from above), consider a sequence of intervals decreasing to \(\{x\}\):
\[
A_n = (x - \frac{1}{n}, x].
\]
Clearly, \(\bigcap_{n=1}^{\infty} A_n = \{x\}\). Since \(\mu(A_1) = 1 \lt \infty\), continuity from above
applies. Since the Lebesgue measure of an interval is its length:
\[
\begin{align*}
\mu(\{x\})
&= \mu(\bigcap_{n=1}^{\infty} A_n) \\\\
&= \lim_{n \to \infty} \mu((x - \frac{1}{n}, x]) \\\\
&= \lim_{n \to \infty} \frac{1}{n} \\\\
&= 0.
\end{align*}
\]
The same conclusion extends to any countable set (like \(\mathbb{Q}\), the rational numbers),
which also has measure zero because \(\mu(\bigcup_i \{x_i\}) = \sum_i \mu(\{x_i\}) = 0\). That fact is
essentially why the "probability" of picking an exact integer from the real line is zero.
Since \(\mu(\mathbb{R}) = \infty\), this measure is not a probability measure.