[**[The Egoroff Theorem for Operator-Valued Measures in Locally Convex Spaces]{}**]{}

[**Ján Haluška and Ondrej Hutník**]{}[^1]

Introduction
============

The classical Egoroff theorem states that almost everywhere convergent sequences of measurable functions on a finite measure space converge almost uniformly, that is, for every $\varepsilon >
0$ the convergence is uniform on a set whose complement has measure less than $\varepsilon$, cf.  [@Halmos].  A necessary and sufficient condition for a sequence of measurable real functions to be almost uniformly convergent is given in [@Bartle2].  In generalizing to functions taking values in more general spaces, some problems appear arising from the fact that the classical relationship between the pointwise convergence and the convergence in measure is not saved.  Namely, any convergence almost everywhere does not imply convergence in measure in general.  For a complete measure space $(T, \Sigma,
\mu)$, and a locally convex space $\bold X$ the following condition may be considered, cf.  [@Goguadze]: [*Let $\mathcal{M}$ be a family of $\bold X$-valued functions defined on $T$. The locally convex space $\bold X$ is said to satisfy the finite Egoroff condition with respect to $\mathcal{M}$ if and only if every sequence in $\mathcal{M}$ which converges almost everywhere to a function, is almost uniformly convergent on every measurable set $A \in \Sigma$ of a finite measure.*]{} 

However, it is not so easy to find such a family of functions $\mathcal{M}$. Thus, E. Wagner and W. Wilczyński proved the following theorem in [@WilWag]. 

If a measurable space $(T, \Sigma, \mu)$ fulfils the countable chain condition, then the convergence $I$-a.e.  is equivalent to the convergence with respect to the $\sigma$-ideal $I$ if and only if $\Sigma/I$ is atomic. 

In the operator-valued measure theory in Banach spaces the pointwise convergence (of sequences) of measurable functions on a set of finite semivariation implies the convergence in (continuous) semivariation of the measure $\bold{m}:\Sigma
\to L(\bold X, \bold Y)$, where $\Sigma$ is a $\sigma$-algebra of subsets of a set $T \neq \emptyset$, and $\bold X, \bold Y$ are Banach spaces, cf.  [@Bartle], and [@Dobrakov].  If $\bold
X$ fails to be metrizable, the relationship between these two convergences is quite unlike the classical situation, cf.  Example after Definition 1.11 in [@Smith-Tucker]. 

It is also well-known that the Egoroff theorem cannot hold for arbitrary nets of measurable functions without some restrictions on measure, net convergence of functions, or class of measurable functions.  For instance, the net of characteristic functions of finite subsets of $[0, 1]$ shows that one needs very special measures (supported by a countable set) to have almost uniform convergence.  In [@Hal91], Definition 1.2, the first author introduced the so called Condition (GB) under which everywhere convergence of net of measurable functions implies convergence of these functions in semivariation on a set of finite variation of measure in locally convex setting, cf.  [@Hal94], Theorem 3.3 (without the assumption of countable chain condition).  This condition concerns families of submeasures and enables to work with nets of measurable functions instead of sequences.  The Condition (GB) is a generalization of Condition (B) which is important for sequences in the classical measure and integration theory, cf.  [@Luzin].  The analogous condition for nets in the classical setting was introduced and investigated by B. F. Goguadze, cf.  [@Goguadze]. 

Recall that Condition (GB) is fulfilled in the case of atomic operator-valued measures, cf.  [@Hal94].  Atomic measures are not of a great interest in the classical theory of measure and integral, because they lead only to considerations of absolutely convergent series.  But when we consider measures with very general range space, e.g. a locally convex space, the situation changes. 

In this paper we prove the Egoroff theorem for atomic operator-valued measures in locally convex topological vector spaces. 

Preliminaries
=============

The description of the theory of locally convex topological vector spaces may be found in [@Jar81].  By a [*net*]{} (with values in a set $S$) we mean a function from $I$ to $S$, where $I$ is a directed partially ordered set.  Throughout this paper $I$ is a directed index set representing direction of a net.  For terminology concerning nets, see [@Keley]. 

Let $T$ be a non-void set and let $\Sigma$ be a $\sigma$-algebra of subsets of $T$. By $2^T$ we denote the potential set of $T$. Let $\bold X$ and $\bold Y$ be two Hausdorff locally convex topological vector spaces over the field $\mathbb{K}$ of all real $\mathbb{R}$ or complex numbers $\mathbb{C}$, with two families of seminorms $P$ and $Q$ defining the topologies on $\bold X$ and $\bold Y$, respectively.  Let $L(\bold X, \bold Y)$ denote the space of all continuous linear operators $L: \bold X\to \bold Y$, and let $\mathbb{N}$ be the set of all natural numbers. 

In what follows $\bold m: \Sigma \to L(\bold X, \bold Y)$ is an operator-valued measure $\sigma$-additive in the strong operator topology of the space $L(\bold X, \bold Y)$, i.e. $\bold
m(\cdot) \bold x: \Sigma \to \bold Y$ is a $\bold Y$-valued vector measure for every $\bold x \in \bold X$.

Let $p \in P$, $q \in Q$, $E \in \Sigma$ and $\bold m: \Sigma \to
L(\bold X, \bold Y)$.

-   The $(p, q)$-[*semivariation of the measure*]{} $\bold m$ is the set function $\hat{\bold m}_{p, q}:\Sigma \to [0,
    \infty ]$, defined as follows: $$\hat{\bold m}_{p, q} (E) =  \sup q \left( \sum_{n = 1}^{N} \bold m(E_n \cap E) \bold x_n \right),$$ where the supremum is taken over all finite disjoint partitions $\{E_n  \in \Sigma; \,E \subset \bigcup_{n=1}^{N} E_n, \,E_n  \cap
    E_m = \emptyset, \,n \neq m,\, m, n = 1, 2, \dots , N \}$ of $E$, and all finite sets $\{ \bold x_n \in \bold X; \,p(\bold x_n) \leq
    1, \, n = 1, 2, \dots , N \}$, $N \in \mathbb{N}$.

-   The $(p, q)$-[*variation of the measure*]{} $\bold m$ is the set function $\bold{var}_{p, q} (\bold m, \cdot): \Sigma \to [0,
    \infty ]$, defined by the equality $$\bold{var}_{p, q}(\bold m, E) = \sup \sum_{n = 1}^{N} q_p \left(\bold m (E_n
    \cap E)\right),$$ where the supremum is taken over all finite disjoint partitions $\{E_n  \in \Sigma; \, E = \bigcup_{n=1}^{N}
    E_n, \, E_n  \cap E_m  = \emptyset, \,n \neq m, \,n, m = 1, 2,
    \dots , N, \, N \in \mathbb{N} \}$ of $E$ and $$q_p(\bold m (F)) =
    \sup_{p(\bold x) \leq 1} q(\bold m(F)\bold x), \quad F \in
    \Sigma.$$

-   The [*inner $(p,q)$-semivariation of the measure*]{} $\bold m$ is the set function $\hat{\bold m}^*_{p,q}: 2^T
    \to [0, \infty ]$, defined as follows: $$\hat{\bold m}^*_{p,q}(E) = \sup_{F \subset E, F \in \Sigma}
    \hat{\bold m}_{p,q}(F), \quad E \in 2^T.$$

Note that $\hat{\bold m}_{p, q}(E) \leq \bold{var}_{p, q}(\bold m,
E)$ for every $q \in Q$, $p \in P$, and $E \in \Sigma$. The following lemma is obvious. 

\[lemma2.2\] For every $p \in P$ and $q \in Q$, the $(p, q)$-(semi)variation of the measure $\bold m$ is a monotone and $\sigma$-additive ($\sigma$-subadditive) set function, and $\bold{var}_{p, q}(\bold
m, \emptyset)=0$, ($\hat{\bold m}_{p, q}(\emptyset)= 0)$.

Let $\bold m: \Sigma \to L(\bold X, \bold
Y)$.

-   The set $E \in \Sigma$ is said [*to be of positive variation of the measure*]{} $\bold m$ if there exist $q \in Q$, $p \in P$, such that $\bold{var}_{p,
    q}(\bold m, E) > 0$.

-   We say that the set $E \in \Sigma$ [*is of finite variation of the measure*]{} $\bold m$ if to every $q \in Q$ there exists $p \in P$, such that $\bold{var}_{p, q} (\bold m, E)<
    +\infty$. We will denote this relation shortly $Q \to_E P,$ or, $q\mapsto_E p$, for $q \in Q$, $p \in P$.

The relation $Q \to_E P$ may be different for different sets $E
\in \Sigma$ of finite variation of the measure $\bold m$.

\[Definition 1.1\] A measure $\bold m: \Sigma \to L(\bold X, \bold Y)$ is said to [*satisfy Condition (GB)*]{} if for every $E \in \Sigma$ of finite and positive variation and every net $(E_i)_{i\in I}$, $E_i
\subset E$, of sets from $\Sigma$ there holds $$\limsup_{i \in I} E_i \neq \emptyset,$$ whenever there exist real numbers $\delta (q, p, E) > 0$, $p \in
P$, $q \in Q$, such that $\hat {\bold m}_{p,q}(E_i) \geq \delta
(q, p, E)$ for every $i \in I$ and every couple $(p, q) \in P
\times Q$ with $q \mapsto_E p$.

Note that the everywhere convergence of a net of measurable functions implies the convergence of these functions in semivariation on a set of finite variation if and only if Condition (GB) is fulfilled.  For instance, the Lebesque measure does not satisfy Condition (GB) on the real line (for arbitrary nets of sets). 

We say that a set $E \in \Sigma$ of positive semivariation of the measure $\bold m$ is an $\hat{\bold m}$-[*atom*]{} if every proper subset $A$ of $E$ is either $\emptyset$ or $A \notin
\Sigma$.

We say that the measure $\bold m$ is [*atomic*]{} if each $E \in
\Sigma$ can be expressed in the form $E=\bigcup_{k = 1}^{\infty}
A_k$, where $A_k, k \in \mathbb{N}$, are $\hat{\bold m}$-atoms. 

In [@Hal94] it is shown that a class of measures satisfying Condition (GB) is non-empty and the following result is proved. 

\[thmatomic\] If $\bold m$ is a (countable) purely atomic operator-valued measure, then $\bold m$ satisfies Condition (GB). 

We say that a function $\bold f: T \to \bold X$ is [*measurable*]{} if $$\{t \in T; \,p(\bold f(t)) \geq  \eta\} \in
\Sigma$$ for every $\eta > 0$ and $p \in P$.

\[Definition 1.2\]A net $(\bold f_i)_{i \in I}$ of measurable functions is said to be [*$\bold m$-almost uniformly convergent*]{} to a measurable function $\bold f$ on $E \in \Sigma$ if for every $\varepsilon >
0$ and every $(p,q) \in P\times Q$ with $q\mapsto_{E} p$ there exist measurable sets $F=E(\varepsilon, p, q)$, such that $$\lim_{i \in I} \|\bold f_i - \bold f\|_{E \setminus
F,p} = 0\quad \textrm{and} \quad \hat{\bold m}_{p, q}(F) <
\varepsilon,$$ where $\|\bold g\|_{G,p}=\sup\limits_{t\in
G}p(\bold g(t))$.

In [@Hal97] the concept of generalized strong continuity of the semivariation of the measure is introduced.  Note that this notion enables development of the concept of an integral with respect to the $L(\bold X, \bold Y)$-valued measure based on the net convergence of simple functions.  For this purpose the notion of the inner semivariation is used for this generalization.  This way we restrict the set of $L(\bold X, \bold Y)$-valued measures which can be taken for such type of integration.  For instance, every atomic measure is generalized strongly continuous.  So, the class of measures with the generalized strongly continuous semivariation is nonempty. 

\[defGS\] We say that the semivariation of the measure $\bold m:
\Sigma \to L(\bold X, \bold Y)$ is [*generalized strongly continuous ([GS]{}-continuous, for short)*]{} if for every set of finite variation $E \in \Sigma$ and every monotone net of sets $(E_i)_{i\in I} \subset T$, $E_i \subset E$, $i \in I$, the following equality $$\lim_{i \in I} \hat{\bold m}_{p,q}^*(E_i) =
\hat{\bold m}_{p,q}^*\left(\lim_{i \in I} E_i\right)$$ holds for every couple $(p, q) \in P \times Q $, such that $q \mapsto_E p$.

The following result connecting the notion of GS-continuity and Condition (GB) was proved in [@Hal97]. 

\[thmGB\] If the semivariation of a measure $\bold m: \Sigma \to L(\bold X,
\bold Y)$ is [GS]{}-continuous, then the measure $\bold m$ satisfies Condition (GB). 

Egoroff theorem for atomic operator-valued measures in locally convex spaces
============================================================================

Before proving the main result of this paper we state the following useful lemma. 

\[lemma\] If $\bold m: \Sigma \to L(\bold X, \bold Y)$ is a (countable) purely atomic measure, then its semivariation is [GS]{}-continuous. 

[[**Proof.    **]{}]{}Let $E \in \Sigma$ be a set of finite and positive variation of the measure $\bold m$. Let $(E_i)_{i \in I}$ be an arbitrary decreasing net of sets from $\Sigma$. Recall that $E_i
\searrow G (\in 2^G)$ if and only if

-   $i \leq j \Rightarrow E_i \supset E_j$, and

-   $\bigcap_{i \in I} E_i = G$.

It is clear that it is enough to consider the case $G =
\emptyset$, because $$E_i \searrow G \Leftrightarrow G \subset
E_i, \quad E_i \setminus G \searrow \emptyset.$$ First, in the case $E_i \in \Sigma$, $i \in I,$ we have $$\label{4}
\lim_{i \in I} \hat{\bold m}^*_{p, q}(E_i) = \lim_{i \in I}
\hat{\bold m}_{p, q}(E_i),$$ and since the family of atoms is at most a countable set, there is $$\lim_{i \in I} E_i = \bigcap_{i \in I} E_i \in \Sigma$$ and, therefore, $$\label{5}
\hat{\bold m}^*_{p,q}\left(\lim_{i \in I} E_i\right) = \hat{\bold
m}_{p, q}\left(\lim_{i \in I} E_i\right)$$ for every $p \in P$, $q \in Q$ such that $q \to_{E} p$.

Now, take an arbitrary set $E \in \Sigma$ of positive and finite variation.  Denote by $\mathcal{A}$ the set of all $\hat{\bold
m}$-atoms, and put $l(i, E) = (\mathcal{A} \cap E) \setminus E_i$, $i \in I$. Clearly $$i \leq j, \,\,i, j \in I \Rightarrow l(i, E)
\subset l(j, E)$$ and there exist atoms $A_n \in \mathcal{A}$, $n\in \mathbb{N}$, such that $l(i, E) = \{ A_1, A_2, \dots , A_n,
\dots\}$.

By Lemma \[lemma2.2\] we have $$\bold{var}_{p,q}(\bold{m},E) = \bold{var}_{p,q}(\bold{m},E_{i}) +
\sum_{A_{n}\in l(i,E)} \bold{var}_{p,q}(\bold{m},A_{n}),$$ for $i\in I$, and $p\in P$, $q\in Q$. Since $$\hat{\bold{m}}_{p,q}(E_{i}) \leq \bold{\bold{var}}_{p,q}(\bold{m},E_{i}), \quad i\in I, p\in P, q\in
Q,$$ there is $$\hat{\bold m}_{p,q}(E_i) \leq \bold{var}_{p,q}(\bold{m},E) -
\sum_{A_n \in l(i, E)} \bold{var}_{p, q}(\bold{m},A_n),$$ where $i\in I$, $p \in P$, $q \in Q$. Since $\bold{var}_{p,q}(\bold{m},E\cap\cdot): \Sigma \to [0,\infty)$ is a finite real measure for every $p\in P$, $q\in Q$ with $q\mapsto_{E}p$, then for every $\varepsilon >0$, $p\in P$, $q\in
Q$ such that $q\mapsto_{E}p$, there exists an index $i_{0}=i_{0}(\varepsilon, p,q,E) \in I$, such that $$\label{6}
\hat{\bold{m}}_{p,q}(E_{i}) < \varepsilon$$ holds for every $i \geq i_{0}$, $i\in I$. Combining (\[4\]), (\[5\]), (\[6\]) and Definition \[defGS\] we see that the assertion is proved for the case when $(E_i)_{i\in I}$ is a decreasing net of sets from $\Sigma$. The other cases of monotone nets of sets may be proved analogously. 

Let now $G \subset T$ be an arbitrary set.  Then there is exactly one (countable) set $F^* = \mathcal{A} \cap G$ with the property: $$\hat{\bold m}^*_{p, q}(G) =
\sup_{F \subset G, F \in \Sigma} \hat{\bold m}_{p, q}(F) =
\hat{\bold m}_{p, q}(F^*), \quad p \in P, q \in Q.$$

The proof for the inner measure and the arbitrary net of subsets $(E_i)_{i \in I}$ goes by the same procedure as in the previous part of proof concerning the set system $\Sigma$. [$\Box\;\;$ ]{}

Observe that Lemma \[lemma\] and Theorem \[thmGB\] imply the statement of Theorem \[thmatomic\]. 

\[Theorem 2.1\] Let $\bold m: \Sigma \to L(\bold X, \bold Y)$ be a purely atomic measure, and let $E \in \Sigma$ be of finite variation of the measure $\bold m$. Let $\bold f: T \to \bold X$ be a measurable function, and $(\bold f_i: T \to \bold X)_{i \in I}$ be a net of measurable functions, such that $$\label{net1}
\lim_{i \in I} p(\bold f_i(t) - \bold f(t)) = 0 \quad \textrm{for
every\,\,\,} t \in E\,\, \textrm{and\,\,\,} p \in P.$$ Then a net $(\bold f_i)_{i \in I}$ of functions $\bold m$-almost uniformly converges to $\bold f$ on $E \in \Sigma$.

[[**Proof.    **]{}]{}We have to prove that for a given $\varepsilon > 0$ and every $q \in Q$, $p \in P$, such that $q \to_E p$ there exist measurable sets $F = E(\varepsilon, p, q) \in \Sigma$, such that $\lim_{i \in I} \| \bold f_i -   \bold f \|_{E \setminus F,p}=0,$ and $\hat{\bold m}_{p, q}(F) < \varepsilon$.

Suppose that (\[net1\]) holds.  For every $m \in \mathbb{N}$, $p
\in P$, and $j \in I$, put [ $$\begin{aligned}
B^p_{m, j} & = & E \cap \left\{ t \in T; \,p(\bold f_i(t)
- \bold f(t)) <  \frac{1}{m}, \quad i \geq j \right\} \\
& = & E \cap \bigcap_{i \geq j} \left\{t \in T; \,p(\bold f_i(t) -
\bold f(t)) < \frac{1}{m}, \quad i \in I \right\}.\end{aligned}$$]{}Since there are countable many of atoms, $B^p_{m, j} \subset
\Sigma$ and $\# B^p_{m, j} =  \aleph_0$. Clearly, if $i, j \in I$ such that $i \leq j$, then $ B^p_{m, i} \subset B^p_{m, j}$ for every $m \in \mathbb{N}$ and $p \in P$. Put $$E^p_m = \bigcup_{j
\in I} B^p_{m, j}.$$ The net $(E^p_m \setminus B^p_{m, i})_{i \in
I}$ clearly tends to void set for every $m \in \mathbb{N}$ and $p
\in P$. Since $\bold m$ is a purely atomic operator-valued measure, then according to Lemma \[lemma\] its semivariation is GS-continuous, and therefore $$\lim_{i \in I} \hat{\bold m}^*_{p, q} (E^p_m \setminus B^p_{m, i}) =
0, \quad q \in Q, \,p \in P,\,\,\textrm{such that } q \mapsto_E
p.$$

Let $\varepsilon > 0$ be given.  To every $p \in P$ and $m \in
\mathbb{N}$ there exists an index $j = j(m, p) \in I$, such that for $q \mapsto _E p$, $$\hat{\bold m}_{p, q}(E^p_m \setminus
B^p_{m, i}) < \varepsilon \cdot \alpha_p \cdot \beta_m,$$ holds for every $i \geq j(m,p)$, where $\{ \alpha_p;\, p \in P \}$ is a summable system of positive numbers in the sense of Moore–Smith and $\{ \beta_m; \,m \in \mathbb{N} \}$ is an absolutely convergent series of positive numbers.  Put $$F = \bigcup_{m \in
\mathbb{N}} \bigcup_{p \in P} \Bigl(E^p_m \setminus B^p_{m, j(m,
p)}\Bigr).$$ So, we have: [ $$\begin{aligned}
\hat{\bold m}^*_{p, q}(F) & = &  \hat{\bold m}^*_{p, q}
\left(\bigcup_{m \in \mathbb{N}} \bigcup_{p \in P}\Bigl(E_m^p
\setminus B^p_{m, j(m, p)}\Bigr) \right) \\ & = & \lim_{ K = \{
p_1, \dots, p_m\}}
\hat{\bold m}_{p, q} \left( \sum_{m = 1}^{\infty} \sum_{p \in K}\Bigl( E_m^p \setminus B^p_{m, j(m, p)} \Bigr) \right) \\
& \leq & \lim_{ K = \{ p_1, \dots, p_m \} } \sum_{m = 1}^{\infty}
\sum_{p \in K} \hat{\bold m}_{p, q} \Bigl(E_m^p \setminus B^p_{m,
j(m, p)}\Bigr)
\\ & < & \varepsilon.\end{aligned}$$]{}

Let us show that the convergence of net of functions $(\bold
f_i)_{i \in I}$ is uniform on $E \setminus F$. Note that $\bigcup_{m = 1}^{\infty} E_m^p = E$ for every $p \in P$. For a given $\eta > 0$ choose an $m_0 \in \mathbb{N}$, such that $\frac{1}{m} < \eta$. Then [ $$\begin{aligned}
E \setminus F & = & E \setminus \bigcup_{m \in
\mathbb{N}} \bigcup_{p \in P}\Bigl(E_m^p \setminus B^p_{m, j(m, p)}\Bigr) \\
& = & \bigcap_{m \in \mathbb{N}} \bigcap_{p \in P} B^p_{m, j(m,
p)} \subset  B^p_{m_0, j(m_0, p)}\end{aligned}$$]{}for every $p \in P$. By definition of the set $B^p_{m_0, j(m_0, p)}$ we have that if $t \in B^p_{m_0, i}$, then $$p(\bold f_i(t) - \bold f(t)) < \eta$$ for every $i \geq j(m_0, p)$. So, (\[net1\]) implies that for every $\eta
> 0$ and $p \in P$ there exists an index $j = j(\eta, p)$, such that for every $i \geq j(\eta, p)$, $i \in I$, there is $$p(\bold f_i(t) - \bold f(t)) < \eta, \,\,\,t \in E \setminus B^p_{m_0, i} \supset E \setminus F,$$ i.e., the net $(\bold f_i)_{i \in I}$ converges uniformly on $E
\setminus F$. The proof is complete. [ $\Box\;\;$ ]{}

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[^1]: [*Mathematics Subject Classification (2000):*]{} Primary 46G10; Secondary 06F20 Operator valued measure, locally convex topological vector spaces, Egoroff theorem, convergence in measure, net convergence of functions.  This paper was partially supported by Grants VEGA 2/0097/08 and VVGS 45/10-11.