Measures

What Does a Measure Record?

A measurable space \((X,\mathcal{F})\) specifies which subsets of \(X\) are available for measurement. A measure adds a way to assign a size to each of those measurable sets. The intended meaning of “size” depends on the setting: it might be a number of points, a length, a probability, or a weighted total. The common structure is that disjoint pieces should have sizes that add, including when there are countably many pieces.

For now, we use that principle to explore examples and consequences; the next tutorial gives the formal definition and develops its hypotheses systematically. In this tutorial, a measure is understood to take values in \([0,\infty]\), to assign size \(0\) to the empty set, and to satisfy countable additivity: whenever \(A_1,A_2,\ldots\) are pairwise disjoint measurable sets, the size of their union is the sum of their sizes. The value \(+\infty\) is allowed, and sums of nonnegative terms are interpreted in the extended nonnegative reals.

The sigma-algebra matters because it is the domain of the measure. It is not generally appropriate to assign a measure to every subset of \(X\): some familiar notions of size, such as length on the real line, are defined only on a specified collection of measurable sets. This is why a measure is considered together with its measurable space, rather than as a size function on an unspecified collection of subsets.

Several Ways to Assign Size

The simplest example is counting. If every subset of \(X\) is measurable, the counting measure assigns to a set its number of elements when it is finite, and \(+\infty\) when it is infinite. This notion of size is useful when each point contributes one unit. Other measures allow points to contribute different amounts, or assign size according to geometric length instead.

A different example concentrates all the size at one point. Fix \(a\in X\). The point mass at \(a\) assigns size \(1\) to a measurable set containing \(a\), and size \(0\) to a measurable set not containing \(a\). For a measurable space, this construction is available whenever \(\{a\}\) is measurable. More generally, multiplying these values by a fixed nonnegative constant gives a point mass of that weight.

Length gives a third interpretation. The standard Lebesgue measure on the Borel subsets of \(\mathbb{R}\) assigns an interval its length: for example, \((2,5)\) has measure \(3\), and a singleton has measure \(0\). This example contrasts with counting measure and point masses: a set can contain infinitely many points but still have finite length, or contain one point and have positive point-mass size.

In probability, the measure of an event is its probability. A probability measure is a measure whose value on the whole sample space is \(1\). For a finite sample space, probabilities can be assigned to individual outcomes and added over the outcomes in an event. This is a weighted version of counting, where the weights need not be equal.

Consequences of the Additivity Principle

Countable additivity immediately controls finite disjoint unions: a finite list can be extended by adding empty sets, whose sizes are zero. It also gives monotonicity: a subset cannot have greater measure than a set containing it. These basic facts are useful in calculations even before more advanced results about measures are developed.

Proof. For finite additivity, append \(A_{m+1}=A_{m+2}=\cdots=\varnothing\) to the finite list. The resulting sequence is pairwise disjoint, so countable additivity gives $$ \mu\left(\bigcup_{j=1}^{m}A_j\right) =\sum_{j=1}^{\infty}\mu(A_j) =\sum_{j=1}^{m}\mu(A_j), $$ because \(\mu(\varnothing)=0\).

For monotonicity, \(A\subseteq B\) implies \(B=A\cup(B\setminus A)\), a disjoint union. Since \(\mathcal{F}\) is closed under relative differences, \(B\setminus A\in\mathcal{F}\). Finite additivity therefore gives $$ \mu(B)=\mu(A)+\mu(B\setminus A)\geq\mu(A), $$ because \(\mu(B\setminus A)\geq0\). This argument remains valid if \(\mu(B)=+\infty\). \(\square\)

The disjointness condition in the additivity statement is essential. If two sets overlap, adding their sizes counts the overlap twice. For example, under length measure, the intervals \((0,2)\) and \((1,3)\) each have length \(2\), but their union is \((0,3)\), of length \(3\), not \(4\). Additivity applies directly to a disjoint decomposition, not to an arbitrary union.

Weighted Counting on a Countable Space

Counting measure gives each point weight \(1\). A useful generalization assigns a nonnegative weight \(w_n\) to each positive integer \(n\). For a subset \(A\) of the positive integers, add the weights of the points in \(A\). If \(A\) is infinite, define the sum as the limit of its increasing partial sums; the value may be \(+\infty\).

Proof. The empty set contains no terms, so \(\mu(\varnothing)=0\). Let \(A_1,A_2,\ldots\) be pairwise disjoint subsets of \(\mathbb{N}\). We show that the weight counted in their union equals the sum of their separate weights.

For each \(N\), only the integers \(1,\ldots,N\) are involved in the partial sum for \(\mu(\bigcup_j A_j)\). Each such integer belongs to at most one \(A_j\), by disjointness. Thus $$ \sum_{\substack{n\leq N\\ n\in\bigcup_j A_j}}w_n =\sum_{j=1}^{\infty}\ \sum_{\substack{n\leq N\\ n\in A_j}}w_n. $$ Only finitely many terms on the right can be nonzero for this fixed \(N\). Letting \(N\) increase, the left side tends to \(\mu(\bigcup_j A_j)\). On the right, all terms are nonnegative, so increasing the finite range of integers collects every weight in every \(A_j\); the resulting limit is \(\sum_j\mu(A_j)\). Equivalently, both sides sum the same nonnegative terms \(w_n\), each exactly once when \(n\) belongs to the union. Therefore $$ \mu\left(\bigcup_{j=1}^{\infty}A_j\right)=\sum_{j=1}^{\infty}\mu(A_j). $$ This proves countable additivity, and hence \(\mu\) is a measure. \(\square\)

Zero, Infinity, and the Choice of Measurable Sets

A set of measure zero need not be empty. Under length measure, a singleton has measure zero; under a point mass concentrated at that singleton, it has measure one. Thus “zero size” is always relative to the chosen measure. Likewise, a set of infinite measure is not necessarily the whole space: under counting measure on \(\mathbb{N}\), the even integers have infinite measure, though they are only part of \(\mathbb{N}\).

It is also important to distinguish a measure from the set of sets on which it is defined. The point-mass example on \(\mathbb{R}\) is defined on Borel sets because its membership condition can be expressed using the measurable singleton \(\{3/2\}\). The weighted construction on \(\mathbb{N}\) uses the power set, so every subset is measurable. Different domains can support different measures and different questions about size.

The central idea is not that there is one universal notion of size. Rather, a measure is a consistent assignment of nonnegative sizes on a specified measurable space. Counting, length, probability, and weighted point masses all fit this pattern while answering different questions. The next tutorial makes the definition precise and examines the conditions that distinguish measures from arbitrary set functions.