From Nested Sets to a Limit of Measures
For an increasing sequence of sets, the measure of the union is the limit of the measures of the individual sets. That statement is useful only when the sets and their measures are actually identified correctly. Counting measure makes the calculation depend on how many points have appeared so far; a Dirac measure depends on whether its one designated point has appeared. In both settings, an explicit description of the union provides a direct check on the limit.
Throughout, an increasing sequence means \(A_1\subseteq A_2\subseteq\cdots\). The sets in each calculation below are measurable in the stated measure space. We use the continuity-from-below theorem established in the earlier material: for any measure space \((X,\mathcal{M},\mu)\), if \(A_n\in\mathcal{M}\) and \(A_n\subseteq A_{n+1}\), then \(\mu(\bigcup_{n=1}^{\infty}A_n)=\lim_{n\to\infty}\mu(A_n)\). No finiteness assumption on \(\mu(X)\) is required; the limit and the measure of the union may both be \(+\infty\).
For counting measure, it is often helpful to count the new points added at each stage. For a Dirac measure, the key question is simpler: at what first index, if any, does the designated point belong to the sets? Since the sequence is increasing, once the point belongs, it remains in every later set.
Counting Measure: Finite Counts and Infinite Limits
Proof. Define \(D_1=A_1\) and \(D_n=A_n\setminus A_{n-1}\) for \(n\ge2\). The sets \(D_n\) are pairwise disjoint: if \(i<j\), then \(D_i\subseteq A_i\subseteq A_{j-1}\), whereas \(D_j\) is disjoint from \(A_{j-1}\). Every element of \(A\) first occurs in some \(A_n\), so it belongs to exactly one \(D_n\). Thus \(A=\bigcup_nD_n\), and \(A_n=\bigcup_{j=1}^nD_j\). By countable additivity of counting measure, \(\mu(A)=\sum_{j=1}^{\infty}\mu(D_j)\) and \(\mu(A_n)=\sum_{j=1}^n\mu(D_j)\). The partial sums of a series of nonnegative terms increase to the series, including when the series is infinite. Therefore \(\mu(A)=\lim_n\mu(A_n)\). \(\square\)
Worked Example: Integers Not Divisible by Three
Let \(X=\mathbb{N}=\{1,2,\ldots\}\) with its power-set sigma-algebra and counting measure \(\mu\). Set \(A_n=\{k\in\mathbb{N}:k\le 2n,\ 3\nmid k\}\). Find \(\mu(A_n)\), the union, and its measure.
The sets increase because increasing \(n\) raises the upper bound \(2n\) and does not change the divisibility condition. Among \(1,\ldots,2n\), precisely \(\lfloor 2n/3\rfloor\) numbers are divisible by three. Hence \(\mu(A_n)=2n-\lfloor 2n/3\rfloor\). For the first four indices, direct substitution gives \(\mu(A_1)=2-\lfloor2/3\rfloor=2\), \(\mu(A_2)=4-\lfloor4/3\rfloor=3\), \(\mu(A_3)=6-\lfloor6/3\rfloor=4\), and \(\mu(A_4)=8-\lfloor8/3\rfloor=6\). These agree with the sets \(\{1,2\}\), \(\{1,2,4\}\), \(\{1,2,4,5\}\), and \(\{1,2,4,5,7,8\}\), respectively.
An integer \(k\) belongs to the union exactly when it is not divisible by three: if \(3\nmid k\), choose \(n\) with \(2n\ge k\); if \(3\mid k\), it is excluded at every stage. Thus \(A=\mathbb{N}\setminus\{3,6,9,\ldots\}\), which is infinite. The measures computed above are unbounded; for example, \(\mu(A_n)\ge n\), since among each pair \(2j-1,2j\) at least one is not divisible by three. Therefore \(\lim_n\mu(A_n)=+\infty=\mu(A)\), as continuity from below requires. Check: the sequence of counts \(2,3,4,6\) is nondecreasing, and the union contains infinitely many integers, such as every positive integer of the form \(3j+1\).
Worked Example: A Finite Set Growing into an Infinite Set
Again take \((\mathbb{N},\mathcal{P}(\mathbb{N}),\mu)\) with counting measure, and let \(B_n=\{1\}\cup\{2k:1\le k\le n\}\). Compute the measures and the measure of the union.
The even numbers \(2,4,\ldots,2n\) are distinct, and none equals \(1\). Consequently, \(B_n\) has exactly \(n+1\) elements and \(\mu(B_n)=n+1\). In particular, \(B_1=\{1,2\}\) has measure \(2\), \(B_2=\{1,2,4\}\) has measure \(3\), and \(B_3=\{1,2,4,6\}\) has measure \(4\). Each \(B_n\) is contained in \(B_{n+1}\).
The union is \(B=\{1\}\cup\{2,4,6,\ldots\}\). It is infinite, so \(\mu(B)=+\infty\). Also \(\lim_n\mu(B_n)=\lim_n(n+1)=+\infty\). The calculation is consistent with continuity from below. Check: the count formula includes the extra point \(1\), which is not among the \(n\) even numbers; omitting it would give \(n\), not the actual measure \(n+1\).
Dirac Measure: The First Index That Captures the Point
Proof. If \(x\) belongs to the union, the set of indices \(n\) with \(x\in A_n\) is a nonempty subset of \(\mathbb{N}\), so it has a least element \(N\). By increasingness, \(x\in A_n\) for every \(n\ge N\); by minimality, it is absent for \(n<N\). The definition of \(\delta_x\) gives the stated values, whose limit is \(1\), equal to the measure of the union. If \(x\) is not in the union, it belongs to no \(A_n\), so all the measures and the measure of the union equal zero. These two cases exhaust the possibilities. \(\square\)
Worked Example: An Endpoint Approaches the Atom
On \((\mathbb{R},\mathcal{B}_{\mathbb{R}},\delta_5)\), where \(\delta_5\) is the Dirac probability measure at \(5\), define \(A_n=[-n,5-1/n]\). Compute the individual measures and the union.
The left endpoints decrease and the right endpoints increase, so \(A_n\subseteq A_{n+1}\). For every positive integer \(n\), \(5-1/n<5\); therefore \(5\notin A_n\) and \(\delta_5(A_n)=0\). The union is \((-\infty,5)\): each point \(y<5\) lies in \([-n,5-1/n]\) for all sufficiently large \(n\), while \(5\) and every point larger than \(5\) lie in none of the sets. Hence \(\delta_5(\bigcup_nA_n)=0=\lim_n\delta_5(A_n)\).
Now define \(C_n=A_n\) for \(n=1,2\), and \(C_n=A_n\cup\{5\}\) for \(n\ge3\). This is still increasing. The first two measures are \(\delta_5(C_1)=0\) and \(\delta_5(C_2)=0\); for every \(n\ge3\), \(\delta_5(C_n)=1\). Its union is \((-\infty,5]\), so \(\delta_5(\bigcup_nC_n)=1=\lim_n\delta_5(C_n)\). Check: the change in measure occurs at \(n=3\) because that is the first index at which \(5\) is explicitly included.
A useful extension covers finitely many atoms at once. Suppose \(x_1,\ldots,x_r\) are distinct measurable points of a measurable space \((X,\mathcal{M})\), and \(c_1,\ldots,c_r\ge0\). The measure \(\nu=\sum_{j=1}^r c_j\delta_{x_j}\) is finite, with total mass \(\sum_{j=1}^r c_j\). For a measurable set \(E\), its measure is \(\nu(E)=\sum_{j=1}^r c_j\chi_E(x_j)\). This formula lets us calculate an increasing-set limit by checking each atom separately.
Proof. Fix \(j\). The sets increase, so either \(x_j\) never belongs to their union, in which case \(\chi_{A_n}(x_j)=0\) for every \(n\), or it first belongs at some index, after which \(\chi_{A_n}(x_j)=1\) at every later index. Thus \(\lim_n\chi_{A_n}(x_j)=\chi_{\cup_nA_n}(x_j)\). Since there are only finitely many indices \(j\), limits commute with this finite sum: explicitly, each of its \(r\) terms is eventually constant, so after the largest of their finitely many transition indices all terms equal their limits. Multiplying by \(c_j\) and summing gives \(\lim_n\nu(A_n)=\sum_{j=1}^r c_j\chi_{\cup_nA_n}(x_j)=\nu(\cup_nA_n)\). If \(r=0\), both sides are zero. \(\square\)
Worked Example: Two Point Masses Enter at Different Stages
On \((\mathbb{R},\mathcal{B}_{\mathbb{R}})\), let \(\nu=2\delta_0+3\delta_2\), a finite measure of total mass \(2+3=5\). Define \(E_n=[-n,1-1/n]\) for \(n=1,2\), and \(E_n=[-n,1-1/n]\cup\{2\}\) for \(n\ge3\). Find the measures and their limit.
The intervals are nested: their left endpoints decrease, and \(1-1/n\) increases with \(n\). The additional point \(2\) enters at \(n=3\) and remains thereafter, so the full sequence is increasing. The point \(0\) belongs to every interval, since \(-n\le0\) and \(1-1/n\ge0\) for every \(n\ge1\). The point \(2\) is absent for \(n=1,2\) and present for \(n\ge3\). Therefore \(\nu(E_1)=2\), \(\nu(E_2)=2\), and \(\nu(E_n)=2+3=5\) for \(n\ge3\).
The union is \((-\infty,1)\cup\{2\}\). Indeed, the increasing intervals have union \((-\infty,1)\), and the added point contributes \(2\). This union contains \(0\) and \(2\), so its measure is \(2+3=5\), agreeing with \(\lim_n\nu(E_n)=5\). Check: the only points with positive \(\nu\)-mass are \(0\) and \(2\); counting other real points in the union does not change this measure.
Common Errors and a Reliable Check
Do not replace an approaching endpoint by its limit. In the Dirac example, \(5-1/n\) approaches \(5\), but \(5-1/n<5\) for every \(n\). A mistaken calculation might say “the right endpoint tends to \(5\), so \(5\in A_n\) and \(\delta_5(A_n)=1\).” That is wrong: membership is tested at each actual index, not at the limiting endpoint. The correction is \(5\notin[-n,5-1/n]\), hence every set has Dirac measure zero and so does their union \((-\infty,5)\).
For counting measure, check both the union and the count. A formula for \(\mu(A_n)\) can be correct while a claimed union is not. Verify that every point in the proposed union appears at some finite stage and that no excluded point appears. When counting a union of pieces, check whether those pieces overlap; add their cardinalities directly only when they are disjoint. Finally, distinguish a sequence of finite measures that grows without bound from a finite limiting value: in the first counting example the exact limit is \(+\infty\), not a large finite number.
These calculations matter because continuity from below gives a dependable route from finite-stage information to the measure of a union. Counting measure records the accumulation of new points, while a Dirac measure records the arrival of its designated point. In either case, state the measure space, verify increasingness, compute the sets’ measures, identify the union, and compare the two sides of the continuity formula.
Check Your Understanding
For each question, state which measure space is being used and justify the set membership or count before applying continuity from below.
- With counting measure on \(\mathbb{N}\), let \(F_n=\{k\in\mathbb{N}:k\le n,\ 2\nmid k\}\). Find \(\mu(F_n)\), describe the union, and compute its measure.
- For \(\delta_3\) on \((\mathbb{R},\mathcal{B}_{\mathbb{R}})\), let \(G_n=(-n,3-1/n]\). Find every \(\delta_3(G_n)\) and the measure of the union.
- Let \(H_n=\{1,\ldots,2n\}\) under counting measure on \(\mathbb{N}\). Compute \(\mu(H_n)\) and the measure of \(\bigcup_nH_n\).
- For \(\nu=4\delta_1+\delta_4\), describe how to calculate \(\nu(A_n)\) for any increasing measurable sequence, using the first index at which each atom is included.
- Give an example of an increasing sequence under a Dirac measure whose measures change from zero to one, and identify the first index at which the change occurs.