Weak convergence of topological measures

Topological measures and deficient topological measures are defined on open and closed subsets of a topological space, generalize regular Borel measures, and correspond to certain non-linear functionals. They lack subadditivity, and many standard techniques of measure theory and functional analysis do not apply to them. Nevertheless, we show that many classical results of probability theory hold for topological and deficient topological measures. In particular, we prove a version of Aleksandrov's Theorem for equivalent definitions of weak convergence of deficient topological measures, as well as a version of Prokhorov's Theorem which ensures that every sequence in a family of topological measures has a weakly convergent subsequence. We define Prokhorov's metric and show that convergence in Prokhorov's metric implies weak convergence of a sequence of deficient topological measures. We also generalize many known results about various dense and nowhere dense subsets of deficient topological measures.

Topological measures and deficient topological measures are defined on open and closed subsets of a topological space, which means that there is no algebraic structure on the domain. They lack subadditivity and other properties typical for measures, and many standard techniques of measure theory and functional analysis do not apply to them. Nevertheless, we show that many classical results of probability theory hold for topological and deficient topological measures. In particular, we prove versions of Aleksandrov's Theorem for equivalent definitions of weak convergence of topological and deficient topological measures as well as a version of Prokhorov's Theorem which ensures that every sequence in a family of topological measures has a weakly convergent subsequence. We define Prokhorov's metric and show that convergence in Prokhorov's metric implies weak convergence of a sequence of deficient topological measures. We also generalize many known results about various dense and nowhere dense subsets of deficient topological measures.
In this paper X is a locally compact space Hausdorff space. By C(X) we denote the set of all real-valued continuous functions on X with the uniform norm, by C 0 (X) the set of continuous functions on X vanishing at infinity, by C c (X) the set of continuous functions with compact support, and by C + 0 (X) the collection of all nonnegative functions from C 0 (X).
When we consider maps into extended real numbers we assume that any such map is not identically ∞.
We denote by E the closure of a set E, and by a union of disjoint sets. A set A ⊆ X is called bounded if A is compact. We denote by id the identity function id(x) = x, and by 1 K the characteristic function of a set K. By supp f we mean {x : f (x) = 0}. We say that Y is dense in Several collections of sets are used often. They include: O(X); C (X); and K (X)the collection of open subsets of X; the collection of closed subsets of X; and the collection of compact subsets of X, respectively. Definition 1. Let X be a topological space and ν be a set function on a family E of subsets of X that contains O(X) ∪ C (X). We say that • ν is compact-finite if ν(K) < ∞ for any K ∈ K (X); • ν is simple if it only assumes values 0 and 1; • a nonnegative set-function ν is finite if ν(X) < ∞; • ν is inner regular (or inner compact regular) if ν(A) = sup{ν(C) : C ⊆ A, C ∈ K (X)} for A ∈ E; • ν is inner closed regular if ν(A) = sup{ν(C) : C ⊆ A, C ∈ C (X)} for A ∈ E; • ν is outer regular if ν(A) = inf{ν(U ) : A ⊆ U, U ∈ O(X)} for A ∈ E. For the following fact see, for example, [14,Chapter XI,6.2] and [8,Lemma 7]. Lemma 3. Let K ⊆ U, K ∈ K (X), U ∈ O(X) in a locally compact space X. Then there exists a set V ∈ O(X) such that C = V is compact and K ⊆ V ⊆ V ⊆ U. If X is also locally connected, and either K or U is connected, then V and C can be chosen to be connected. Definition 4. A deficient topological measure on a locally compact space X is a set function ν : C (X) ∪ O(X) −→ [0, ∞] which is finitely additive on compact sets, inner compact regular, and outer regular, i.e. : Clearly, for a closed set F , ν(F ) = ∞ iff ν(U ) = ∞ for every open set U containing F . If two deficient topological measures agree on compact sets (or on open sets) then they coincide.
satisfying the following conditions: By DTM(X) and TM(X) we denote, respectively, the collections of all finite deficient topological measures and all finite topological measures on X.
The following two theorems from [11,Section 4] give criteria for a deficient topological measure to be a topological measure or a measure. Theorem 6. Let X be compact, and ν a deficient topological measure. The following are equivalent: Let X be locally compact, and ν a deficient topological measure. The following are equivalent: Theorem 7. Let µ be a deficient topological measure on a locally compact space X. The following are equivalent: The inclusions follow from the definitions. When X is compact, there are examples of topological measures that are not measures and of deficient topological measures that are not topological measures in numerous papers, beginning with [2], [15], and [24]. When X is locally compact, see [7], Sections 5 and 6 in [11], and Section 9 in [8] for more information on proper inclusion in (1), criteria for a deficient topological measure to be a measure from M (X), and various examples.
Remark 9. In [11, Section 3] we show that a deficient topological measure ν is τ -smooth on compact sets (i.e. if a net K α ց K , where K α , K ∈ K (X) then µ(K α ) → µ(K)), and also In particular, a deficient topological measure is additive on open sets. A deficient topological measure ν is also superadditive, i.e. if t∈T A t ⊆ A, where A t , A ∈ O(X) ∪ C (X), and at most one of the closed sets (if there are any) is not compact, then ν(A) ≥ t∈T ν(A t ). If F ∈ C (X) and C ∈ K (X) are disjoint, then ν(F ) + ν(C) = ν(F ⊔ C). One may consult [11] for more properties of deficient topological measures on locally compact spaces.
A real-valued map ρ on C 0 (X) is a quasi-linear functional (or a positive quasi-linear functional) if
(I) Given a finite deficient topological measure µ on a locally compact space X and f ∈ C b (X), define functions on R: Let r be the Lebesque-Stieltjes measure associated with −R 1 , a regular Borel measure on R. We define a functional on C 0 (X): We call the functional R a quasi-integral (with respect to a deficient topological measure µ) and write: (II) Functional R is not linear, but p-conic quasi-linear. By [9, Lemma 7.7, Lemma 7.10, Lemma 3.6, Lemma 7.12] we also have: (assuming at most one of ∞, −∞) and |ρ(0)| < ∞ is called a d-functional if on nonnegative functions it is positive-homogeneous, monotone, and orthogonally additive, i.e. for f, g ∈ D(ρ) (the domain of ρ) we have: Let ρ be a d-functional with C + c (X) ⊆ D(ρ) ⊆ C b (X). In particular, we may take functional R on C + 0 (X) from part (I). The corresponding deficient topological measure µ = µ ρ is given as follows: If given a p-conic quasi-linear functional R we obtain µ = µ R , and then construct from µ functional R µ as in part (I), then R = R µ . If given a finite deficient topological measure µ, we obtain R, and then µ R , then µ = µ R .
The basic neighborhoods for the weak topology have the form Let µ α be a net in DTM(X), µ ∈ DTM(X). The net µ α converges weakly to µ (and we i.e. f dµ α → f dµ for every f ∈ C + 0 (X). By [9,Theorem 8.7], DTM(X) with weak convergence is homeomorphic to Φ + (C + 0 (X)) with pointwise convergence, and TM(X) is homeomorphic to the space of quasi-linear maps with pointwise convergence.

Remark 14.
Our definition of weak convergence corresponds to one used in probability theory. It is the same as a functional analytical definition of wk * convergence on DTM(X) (respectively, on TM(X)), which is justified by the fact that this topology agrees with the weak * topology induced by p-conic quasi-linear functionals (respectively, quasi-linear functionals). In many papers the term "wk * -topology is used.
Remark 16. In probability theory, with µ a measure, a set A is called a µ-continuity set if µ(∂A) = 0. If µ is a measure (or µ is a topological measure and A is compact) this definition is equivalent to Definition 15. If µ is a deficient topological measure, then by superadditivity µ(A) ≥ µ(A o ) + µ(∂A), so for any µ-continuity set A we have µ(∂A) = 0.
We have the following generalizations of Aleksandrov's well-known theorem for weak convergence of measures. (Aleksandrov's Theorem is often incorrectly called the "Portmanteau theorem", a usage apparently deliberately started by Billingsley, who in [4] cited a paper of the non-existent mathematician Jean-Pierre Portmanteau, "published" in a non-existent issue of the Annals of non-existent university; see [21, p.130] and [23, p.313].) This theorem gives equivalent definitions of weak convergence.
Theorem 19. The weak topology on DTM(X) is given by basic neighborhoods of the form Proof. The weak topology is the topology τ N given by basic neighborhoods of the form (3). It is easy to see that the sets W (ν, U 1 , . . . , U n , C 1 , . . . , C m , ǫ) are basic neighborhoods for some topology τ W on DTM(X). Consider a basic neighborhood W (ν, U, C, ǫ). Given ǫ > 0, by part (III) of Remark 12 choose f, g ∈ C c (X) such that suppf ⊆ U, g ≥ 1 K and Let µ ∈ N (ν, f, g, ǫ/2) as in (3). We have: Therefore, N (ν, f, g, ǫ/2) ⊆ W (ν, U, C, ǫ). We see that τ W ⊆ τ N , i.e. τ W is a coarser topology then τ N . If µ α → µ in the topology τ W then it is easy to see that lim inf µ α (U ) ≥ µ(U ) for any open set U , and that lim sup µ α (K) ≥ µ(K) for any compact set K. By Theorem 17 f dµ α → f dµ for every f ∈ C + 0 (X). The weak topology τ N is the coarsest topology with this property, thus, τ N = τ W .

Theorem 20. The space DTM(X) is Hausdorff and locally convex. Every set of the form {µ ∈
which is a contradiction.

PROKHOROV'S THEOREM FOR TOPOLOGICAL MEASURES
In this section we show that several classical results of probability theory hold for deficient topological measures or topological measures.
Lemma 21. If each sequence {µ n i } of {µ n }, where µ n are deficient topological measures, contains a further subsequence {µ n i j } such that µ n i j converges weakly to a deficient topological measure µ, then µ n converges weakly to µ.
Proof. If µ n does not converge weakly to µ, then there is f ∈ C + 0 (X) such that | f dµ n i − f dµ| ≥ ǫ for some ǫ > 0 and all µ n i in some subsequence. But then no subsequence of {µ n i } can converge weakly to µ. Proof. Suppose X is a separable metric space. By Urysohn's metrization theorem (see [17, p.125]) X can be topologically embedded in a countable product of unit intervals. Consequently, there exists an equivalent totally bounded metrization on X. We will consider this metric on X. From [20,Lemma 6

We clearly have
Let Y be a countable product of R. Define a map T : P −→ Y as in Theorem 20, i.e. T (µ) = ( f 1 dµ, f 2 dµ, . . .). We will show that T is a homeomorphism on P . First, T is 1 − 1.
(If T (µ) = T (ν) then f i dµ = f i dν for all i, and, hence, f dµ = f dν for all f ∈ C + 0 (X). By Remark 12, µ = ν.) Second, T and T −1 are continuous, as in the proof of Theorem 20. Since Y is a separable metric space, and P is homeomorphic to a subset of Y , it follows that P is a separable metric space.
Conversely, suppose P is a separable metric space. By Lemma 22 X is homeomorphic to D = {δ x : x ∈ X}. D is a separable metric space, and then so is X. Proposition 25. Suppose X is locally compact. If a sequence (µ n ) ∈ DTM(X) is weakly fundamental (i.e. f dµ n is a fundamental sequence for each f ∈ C + 0 (X)) then it is uniformly bounded in variation.
Proof. If not, then there is a subsequence (µ n k ) such that µ n k > k2 k for each k; and by part (III) of Remark 12 there are functions f n k ∈ C c (X), 0 ≤ f n k ≤ 1 such that X f n k dµ n k > k2 k .
Then the function f = ∞ k=1 fn k 2 k ∈ C + 0 (X), 0 ≤ f ≤ 1, and X f dµ n k ≥ k for each k. This contradicts the fact that the sequence ( f dµ n ) is Cauchy, hence, bounded. Proof. If not, then there is a sequence µ n ⊆ M such that µ n > n for every natural n. Let m n k be its weakly convergent subsequence. Then m n k > n k , while by Proposition 25 this subsequence must be uniformly bounded in variation. Proof. Suppose M is not uniformly tight. Then there exists ǫ > 0 such that for every compact K one can find µ K ∈ M with Take µ 1 to be any topological measure with µ 1 > ǫ, and let K 1 ∈ K (X) be such that µ(K 1 ) > ǫ. Then by Lemma 3 there is V 1 ∈ O(X) with compact closure such that K 1 ⊆ V 1 and so µ 1 (V 1 ) > ǫ. By (4) find µ 2 satisfying µ 2 (X \ V 1 ) > ǫ, and let K 2 ∈ K (X) be such that K 2 ⊆ X \V 1 and µ(K 2 ) > ǫ. Find V 2 ∈ O(X) with compact closure such that K 2 ⊆ V 2 ⊆ V 2 ⊆ X \V 1 , so µ 2 (V 2 ) > ǫ. Find a topological measure µ 3 with µ 3 (X \ (V 1 ⊔ V 2 ) > ǫ, and so on. By induction we find a a sequence of compact sets K j , a sequence of open sets V j with compact closure, and a sequence of topological measures µ j ∈ M with the following properties:

By part (III) of Remark 12 find functions
By our assumption the sequence (µ j ) contains a weakly convergent subsequence. For notational simplicity, assume that (µ j ) is weakly convergent.
By Lemma 26 we may assume that M is uniformly bounded in variation by M . We let Then a n := (a 1 n , a 2 n , . . . , ) belongs to l 1 , because for each m ∈ N, f 1 · f 2 · . . . · f m = 0, f 1 + . . . + f m ∈ C c (X), 0 ≤ f 1 + . . . + f m ≤ 1, and so by part (III) of Remark 12 each partial sum the sequence (b n ) is bounded, and we may chose a convergent subsequence. To simplify notations, we assume that (b n ) itself converges. Let λ = (λ i ) ∈ l ∞ . Since | λ, a n | ≤ λ ∞ a n 1 ≤ λ M, we see that the sequence of inner products λ, a n is bounded, hence, contains a convergent subsequence. Again, for notational simplicity we assume the sequence itself converges.
By [5, Lemma 1.3.7] the sequence (a n ) converges in l 1 −norm. Then lim n→∞ a n n = 0, which contradicts our choice of f n .
Lemma 28. Let X be locally compact. If (µ n ) is a weakly fundamental sequence of finite deficient topological measures which is also uniformly bounded in variation, then µ n converges weakly to some finite deficient topological measure µ.
Proof. Consider functional L on C + 0 (X) defined as L(f ) = lim n X f dµ n . It is easy to check that L is a p-conic quasi-linear functional. Say, (µ n ) is uniformly bounded in variation by M .
Theorem 29. Suppose X is a locally compact space such that C + 0 (X) is separable. Then every uniformly bonded in variation sequence of finite topological measures has a subsequence which is weakly fundamental.
Proof. Suppose (µ n ) ∈ DTM(X) and µ n ≤ M for each n. Let g ∈ C + 0 (X), so 0 ≤ g ≤ b for some b. Each of the functions R 2,µn,g (t) is monotone and bounded above by M on [0, b]. By the Helly-Bray theorem (see [5,Theorem 1.4.6]), there is pointwise convergent subsequence R 2,µn i ,g .
Then the sequence of integrals X g dµ n i = b 0 R 2,µn i ,g (t)dt converges, hence, is fundamental. If G is a countable dense set in C + 0 (X), we pick a first subsequence of (n i ) such that ( X g 1 dµ n i ) is fundamental for the first function g 1 ∈ G, then we choose a further subsequence (n i j ) for which ( X g 2 dµ n i j ) is fundamental for the function g 2 ∈ G, and so on. By diagonal process we obtain a subsequence of (µ n ) for which the sequence of integrals is fundamental for each g ∈ G. For notational simplicity, let us assume that (µ n ) is such a subsequence, i.e. ( X g dµ n ) is fundamental for each function g ∈ G.
For arbitrary f ∈ C + 0 (X) and ǫ > 0 choose g ∈ G such that f − g ≤ ǫ and n 0 such that | X g dµ n − X g dµ i | < ǫ for n, i ≥ n 0 . Then using [10, Corollary 53] we have: and the sequence of integrals ( X f dµ n ) is fundamental. Thus, (µ n ) is weakly fundamental.
Remark 30. If X is a locally compact Hausdorff space which is second countable or satisfies any of the other equivalent conditions of [16,Theorem 5.3,p.29], thenX, the Aleksandrov one-point compactification of X, is a compact metrizable (hence, a second countable) space. Then C(X) is separable, and C 0 (X) is also separable as as a subspace of a separable metric space.
For topological measures we have the following version of Prokhorov's well-known theorem.
Lemma 32. d P is a metric on DTM(X).

DENSITY THEOREMS
Definition 34. A deficient topological measure ν is called proper if from m ≤ ν, where m is a Radon measure it follows that m = 0.
Remark 35. From [12,Theorem 4.3] it follows that a finite deficient topological measure can be written as a sum of a finite Radon measure and a proper finite deficient topological measure. The sum of two proper deficient topological measures is proper (see [12,Theorem 4.8

]).
A finite Radon measure on a compact space is a regular Borel measure, so our definition (which is given in [12]) of a proper deficient topological measure coincides with definitions in papers prior to [12].
In what follows, pDTM(X) and pTM(X) denote, respectively, the family of proper finite deficient topological measures and the family of finite topological measures.
Let X be a locally compact non-compact space. A set A is called solid if A is connected, and X \ A has only unbounded connected components. When X is compact, a set is called solid if it and its complement are both connected. For a compact space X we define a certain topological characteristic, genus. See [3] for more information about genus g of the space. A compact space has genus 0 iff any finite union of disjoint closed solid sets has a connected complement. Intuitively, X does not have holes or loops. In the case where X is locally path connected, g = 0 if the fundamental group π 1 (X) is finite (in particular, if X is simply connected). Knudsen [18] was able to show that if H 1 (X) = 0 then g(X) = 0, and in the case of CW-complexes the converse also holds.
Remark 36. From Theorem 6 it is easy to see that if µ, ν are deficient topological measures, and ν is not a topological measure, then µ + ν is a deficient topological measure which is not a topological measure. Proof. We shall prove the first part; the proof of the second part is similar, but simpler.
(A) We shall show the first implication. Any measure is approximated by convex combinations of point-masses, so by assumption, it is approximated by convex combinations of proper simple deficient topological measures that are not topological measures. By Remark 35 and Remark 36 the latter combinations are in pDTM(X) \ TM(X). (B) (pDTM(X) \ TM(X) is dense in M(X)) =⇒ (pDTM(X) \ TM(X) is dense in DTM(X)\TM(X)): Suppose µ ∈ DTM(X)\TM(X). By Remark 35 write µ = m+ µ ′ , where µ ′ is a proper deficient topological measure, and m is a measure from M(X). By assumption, m is approximated by ν ∈ pDTM(X) \ TM(X). Then µ is approximated by ν + µ ′ , where by Remark 35 and Remark 36 ν + µ ′ is in pDTM(X) \ TM(X).
is dense in M(X)): Suppose to the contrary that there exists a measure m ∈ M(X) and its neighborhood N which contains no elements of pDTM(X) \ pTM(X). Take λ ∈ DTM(X) \ TM(X). Then for any deficient topological measure ν ∈ N we see that λ + ν is a deficient topological measure that is not a topological measure and is not proper. Thus, a neighborhood λ + N ⊆ DTM(X) \ TM(X) contains no elements of pDTM(X) \ TM(X), which contradicts the assumption. converges weakly to δ a .
Remark 39. Among spaces that satisfy the condition of the previous theorem are: non-singleton locally compact spaces that are locally connected or weekly locally connected; manifolds; CW complexes.
Theorem 40. Suppose X is a non-singleton connected, locally connected, locally compact space with no cut points and such that the Aleksandrov one-point compactification of X has genus 0.
Then pTM(X) is dense in TM(X), and pDTM(X) is dense in DTM(X).
Proof. We shall give the proof for the case when X is not compact. (When X is compact the proof is similar but simpler; also, one may use [26,Theorem 4.9].) We shall show that proper simple topological measures are dense in the set of simple measures, and the statements will follow from part (2) of Theorem 37. Let δ a be a point-mass. It is enough to show that a neighborhood of the form W (δ a , U, C, ǫ) as in Theorem 19 contains a simple proper topological measure.
The remaining three cases are easy. For example, if a ∈ U, a ∈ C then λ as above will do.
µ i is a finite deficient topological measure. If each µ i is a topological measure, then µ is a finite topological measure.
It is easy to see that µ is finitely additive on compact sets. For ǫ > 0 let j be such that ∞ i=j+1 µ i (X) < ǫ, and let λ = j i=1 µ i . Then λ is a finite deficient topological measure. For U ∈ O(X) there exists K ∈ K (X) such that λ(U ) < λ(K) + ǫ. Then µ(U ) < λ(U ) + ǫ < λ(K) + 2ǫ < µ(K) + 2ǫ, and the inner regularity of µ follows. Similarly, µ is outer regular. Thus, µ is a deficient topological measure; clearly, µ is finite. If each µ i is a topological measure, it is easy to check additivity of µ on O(X) ∪ K (X), so condition (TM1) of Definition 5 holds, and µ is a topological measure.
Lemma 42. Suppose X is locally compact, ∞ i=1 µ i (X) < ∞ where each µ i is a proper deficient topological measure (respectively, a proper topological measure). Then µ = ∞ i=1 µ i is a finite proper deficient topological measure (respectively, a finite proper topological measure).
Proof. By Lemma 41 µ is a finite deficient topological measure (respectively, a finite topological measure). We need to show that µ is proper. By Remark 35 write µ = m + µ ′ , where m is a finite Radon measure and µ ′ is a proper deficient topological measure. We shall show that m = 0.
Let K ∈ K (X). For ǫ > 0 let N be such that ∞ i=N +1 µ i (X) < ǫ, and let µ N = N i=1 µ i . By Remark 35 µ N is a proper deficient topological measure. By [12,Theorem 4.5] there are compact sets K 1 , . . . , K n such that K = ∪K j and n j=1 µ N (K j ) < ǫ. Let E 1 , . . . , E n be disjoint Borel sets such that E j ⊆ K j and n j=1 E i = n j=1 K j . Since m is finite, outer regularity of m is equivalent to inner closed regularity of m. Find disjoint sets C j , C j ⊆ E j ⊆ K j , j = 1, . . . , n such that C j are closed (hence, compact) and m(C j ) > m(E j ) − ǫ n . Then It follows that m(K) = 0 for any K ∈ K (X). Thus, m = 0, and µ is proper.
Proof. Note that each X i is a locally compact space with respect to the subspace topology. We shall prove the first part. Let m ∈ M(X). We shall show that every neighborhood W of m as in Theorem 19 contains a proper deficient topological measure. To simplify notation, we consider It is easy to see that m i ∈ M(X i ).
Let ν i be the extension of λ i to O(X)∪C (X) given by ν i (A) = λ i (A∩X i ) for A ∈ O(X)∪C (X).
It is easy to see that ν i is a deficient topological measure, and ν i (X) = λ i (X i ) < ∞. Since λ i is proper, by [12,Theorem 4.5] given δ > 0 there are sets of the form V j ∩ X i , V j ∈ O(X), j = 1, . . . , n such that they cover X i and n j=1 λ i (V j ∩ X i ) < δ. Then open sets V 1 , . . . , V n , X \ X i cover X and n j=1 ν i (V j ) + ν i (X \ X i ) = n j=1 λ i (V j ∩ X i ) < δ, and so ν i is proper. Thus, ν i ∈ pDTM(X) by [12,Theorem 4.5].
The proof of the second part is the same, taking into account that λ i , ν i , ν are proper topological measures.
where each X i as in Theorem 40. Then pTM(X) is dense in TM(X), and pDTM(X) is dense in DTM(X).
Proof. By part 2 of Theorem 37 it is enough to show that pTM(X) is dense in M(X). By Theorem 40, pTM(X i ) is dense in M(X i ) for each i, and we apply part 2 of Theorem 43.
Remark 45. In Corollary 44 one may take, for example, a compact n-manifold, n ≥ 2 as X, or X that is covered by countably many sets homeomorphic to balls B n with varying n ≥ 2.
Lemma 46. TM(X) is a closed subset of DTM(X), and M(X) is a closed subset of DTM(X).
Proof. By Remark 12 µ ∈ TM(X) iff ρ is a quasi-linear functional on C 0 (X), and µ ∈ M(X) iff ρ is a linear functional on C 0 (X), where ρ(f ) = R µ (f + ) − R µ (f − ). Using basic open sets in Definition 13 it is easy to check that TM(X) is a closed subset of DTM(X), and M(X) is a closed subset of DTM(X).
Theorem 47. Suppose X is locally compact. The following are equivalent: (1) M(X) is nowhere dense in DTM(X) (or in TM(X).) (2) There exists a finite deficient topological measure (respectively, a finite topological measure) that is not a measure. Then each µ n is a deficient topological measure that is not a measure, and µ n =⇒ m by Theorem 17. Thus, DTM(X)\M(X) is dense in M(X), and since M(X) is a closed subset of DTM(X), we see that M(X) is nowhere dense in DTM(X). The proof for topological measures is similar.
Corollary 48. Suppose X is locally compact. If X contains a non-singleton compact connected set, then M(X) is nowhere dense in DTM(X). If X contains an open (or closed) locally connected, connected, non-singleton subset whose Aleksandrov one-point compactification has genus 0 then M(X) is nowhere dense in TM(X).
Proof. Use part (2) of Theorem 47. For the first statement, as an example of a finite deficient topological measure that is not a topological measure (hence, not a measure) one may use [11, Example 46], For the second statement, as an example of a finite topological measure that is not a measure one may take [8, Example 61].
The proof of the next Theorem and Corollary are similar to the proof of Theorem 47 and Corollary 48.
Theorem 49. Suppose X is locally compact. The following are equivalent: (1) TM(X) is nowhere dense in DTM(X).
(2) There exists a finite deficient topological measure that is not a topological measure.
(3) There exists a nonzero finite proper deficient topological measure that is not a topological measure.
Corollary 50. If a locally compact space X contains a non-singleton compact connected set, then TM(X) is nowhere dense in DTM(X).
Remark 51. When the space is compact, the equivalence of the first two conditions in Theorem 17 and of first three conditions in Theorem 18 was first given in [26,Corollary 4.4,4.5]. When X is compact Theorem 19 was proved in [26], but the method there does not work for a locally compact non-compact space, as the set f −1 ([0, ∞)) = X is not compact. Theorem 20 generalizes results from several papers, including [1], [13], and [26]. Theorem