Let ${C}_{b}{\left(X,E\right)}_{\beta}^{\u2033}$ denote the bidual of (*C*_{b}(*X*, *E*), *β*). Since *β*-bounded subsets of *C*_{b}(*X*, *E*) are *τ*_{u}-bounded, the strong topology $\beta \left({C}_{b}{\left(X,E\right)}_{\beta}^{\prime},{C}_{b}\left(X,E\right)\right)$ in ${C}_{b}{\left(X,E\right)}_{\beta}^{\prime}$ coincides with the norm topology in *C*_{b}(*X*, *E*)′ restricted to ${C}_{b}{\left(X,E\right)}_{\beta}^{\prime}$. Hence ${C}_{b}{\left(X,E\right)}_{\beta}^{\u2033}={\left({C}_{b}{\left(X,E\right)}_{\beta}^{\prime},\Vert \text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\Vert \right)}^{\prime}$.

Assume that *T* : *C*_{b}(*X*, *E*) → *F* is a (*β*, *||* · *||*_{F})-continuous operator. Let ${T}^{\prime}:{F}^{\prime}\to {C}_{b}{\left(X,E\right)}_{\beta}^{\prime}$ and ${T}^{\u2033}:{C}_{b}{\left(X,E\right)}_{\beta}^{\u2033}\to {F}^{\u2033}$ stand for the conjugate and the biconjugate operators of *T*, respectively, i.e., *T′*(*y′*) := *y′ ◦ T* for *y*′ ∈ *F'*, and *T"*(Ψ)(*y*′) := Ψ(*y*′ ◦ *T*) for $\text{\Psi}\in {C}_{b}{\left(X,E\right)}_{\beta}^{\u2033}\left(={\left({C}_{b}{\left(X,E\right)}_{\beta}^{\prime},\Vert \text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\Vert \right)}^{\prime}\right)$ and *y*′ ∈ *F′*.

Then one can embed *B*(*𝓑o*, *E*) into ${C}_{b}{\left(X,E\right)}_{\beta}^{\u2033}$ by the mapping $\pi :B\left(\text{\U0001d4d1}o,E\right)\to {C}_{b}{\left(X,E\right)}_{\beta}^{\u2033}$, where for *g* ∈ *B*(*𝓑o*, *E*),

$$\pi \left(g\right)\left({\text{\Phi}}_{\mu}\right)=\left(I\right){\displaystyle \underset{X}{\int}g\text{\hspace{0.17em}}d\mu}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\mu \in M\left(X,{E}^{\prime}\right),$$and (*I*) *∫*_{X} g dμ denotes the so-called *immediate integral* (see [1, 2]). Then

$$\left|\pi \left(g\right)\left({\text{\Phi}}_{\mu}\right)\right|=\left|\left(I\right){\displaystyle \underset{X}{\int}g\text{\hspace{0.17em}}d\mu}\right|\le \Vert g\Vert \text{\hspace{0.17em}}\text{\hspace{0.17em}}\left|\mu \right|\left(X\right)=\Vert g\Vert \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Vert {\text{\Phi}}_{\mu}\Vert $$and hence *π* is bounded and ||*π*(*g*)|| ≤ ||g||. One can easily show that

$$\Vert {T}^{\prime}\left({y}^{\prime}\right)\Vert \le \Vert T\Vert \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Vert {y}^{\prime}\Vert}_{{F}^{\prime}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Vert {T}^{\u2033}\left(\text{\Psi}\right)\Vert}_{{F}^{\u2033}}\le \Vert T\Vert \text{\hspace{0.17em}}\text{\hspace{0.17em}}\Vert \text{\Psi}\Vert ,$$where $\Vert \text{\Psi}\Vert =\mathrm{sup}\left\{\left|\text{\Psi}\left(\text{\Phi}\right)\right|:\text{\Phi}\in {C}_{b}{\left(X,E\right)}_{\beta}^{\prime},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\Vert \text{\Phi}\Vert \le 1\right\}$. Then *T* and *T″* are bounded operators, and we define a bounded operator by:

$$\widehat{T}:={T}^{\u2033}\circ \pi :B\left(\text{\U0001d4d1}o,E\right)\to {F}^{\u2033}.$$Define a measure *m* : *𝓑o* → 𝓛(*E*, *F"*) (called the *representing measure* of *T*) by

$$m\left(A\right)\left(x\right):=\widehat{T}\left({\mathrm{\U0001d7d9}}_{A}\otimes x\right)\text{for}\text{\hspace{0.17em}}A\in \text{\U0001d4d1}o,x\in E.$$For each *y′* ∈ *F*′, let

$${m}_{{y}^{\prime}}\left(A\right)\left(x\right):=\left(m\left(A\right)\left(x\right)\right)\left({y}^{\prime}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{\hspace{0.17em}}A\in \text{\U0001d4d1}o,x\in E.$$Then *m*_{y}_{′} *: 𝓑o* → *E′* and by [1, § 4, Proposition 5] for *A* ∈ 𝓑*o* we have,

$$\tilde{m}\left(A\right)=\mathrm{sup}\left\{\left|{m}_{{y}^{\prime}}\right|\left(A\right):{y}^{\prime}\in {B}_{{F}^{\prime}}\right\}.$$Then $\widehat{T}$ admits an integral representation by the so-called *immediate integral* (*I*) *∫*_{X} g dm, developed by Foias and Singer [18] and Dinculeanu [1, § 6], [2, § 1], that is,

$$\widehat{T}\left(g\right)=\left(I\right){\displaystyle \underset{X}{\int}g\text{\hspace{0.17em}}dm}:=\underset{n}{\mathrm{lim}}\left(I\right){\displaystyle \underset{X}{\int}{s}_{n}\text{\hspace{0.17em}}d\text{\hspace{0.17em}}m}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}g\in B\left(\text{\U0001d4d1}o,E\right),$$where (*s*_{n}) is a sequence in *𝓢*(*𝓑o*, *E*) such that *||s*_{n} – g || → 0. Then for *y′* ∈ *F′*,

$$\widehat{T}\left(g\right)\left({y}^{\prime}\right)=\left(\left(I\right){\displaystyle \underset{X}{\int}g\text{\hspace{0.17em}}d\text{\hspace{0.17em}}m}\right)\left({y}^{\prime}\right)=\left(I\right){\displaystyle \underset{X}{\int}g\text{\hspace{0.17em}}d\text{\hspace{0.17em}}{m}_{{y}^{\prime}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{\hspace{0.17em}}g\in B\left(\text{\U0001d4d1}o,E\right).}$$Let *i*_{F} : *F* → *F*″ denote the canonical embedding, i.e., *i*_{F}(*y*)(*y′*) *= y′*(*y*) for *y* ∈ *F*, *y′* ∈ *F′*. Moreover, let *j*_{F} : *i*_{F}(*F*) → *F* stand for the left inverse of *i*_{F}, that is, *j*_{F} ◦ i_{F} = *id*_{F}.

From the general properties of the operator $\widehat{T}$ for *h* ∈ *C*_{rc}(*X*, *E*) we have,

$$\widehat{T}\left({C}_{rc}\left(X,E\right)\right)\subset {i}_{F}\left(F\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}T\left(h\right)={j}_{F}\left(\widehat{T}\left(h\right)\right)={j}_{F}\left(\left(I\right){\displaystyle \underset{X}{\int}h\text{\hspace{0.17em}}d\text{\hspace{0.17em}}m}\right).$$(5)Hence for each *y′* ∈ *F*′, we get

$${y}^{\prime}\left(T\left(h\right)\right)=\widehat{T}\left(h\right)\left({y}^{\prime}\right)=\left(I\right){\displaystyle \underset{X}{\int}h\text{\hspace{0.17em}}d\text{\hspace{0.17em}}{m}_{{y}^{\prime}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{\hspace{0.17em}}h\in {C}_{rc}\left(X,E\right).}$$(6)By *M*(*X*, *𝓛*(*E*, *F″*)) we denote the space of all measures *m* : *𝓑o* → 𝓛(*E*, *F″*) such that $\tilde{m}$(*X*) < ∞ and for each *y′* ∈ *F′*, *m*_{y}_{′} ∈ *M*(*X*, *E′*).

*We say that m* ∈ *M*(*X*, *𝓛*(*E*, *F″*)) *has the tight semivariation if for every ∊* > 0 *there exists K* ∈ *𝓚*(*X*) *such that* $\tilde{m}$(*X \ K*) ≤ *∊*.

Now we can state the following Riesz representation theorem for operators on *C*_{b}(*X*, *E*).

*Assume that T* : *C*_{b}(*X*, *E*) → *F is a* (*β*, || · ||_{F})*-continuous operator and m is its representing measure. Then the following statements hold*:

*m* ∈*M*(*X*, *𝓛*(*E*, *F″*)).

*For each y*′ ∈ *F′*, *y′*(*T*(*f*)) *= ∫*_{X} f dm_{y′} for f ∈ *C*_{b}(*X*, *E*).

*The mapping F'* ∍ *y*′ ↦ *m*_{y}_{′} ∈ *M*(*X*, *E′*) *is* (*σ*(*F'*, *F*), *σ*(*M*(*X*, *E′*), *C*_{b}(*X*, *E*)))*-continuous*.

*m has the tight semivariation*.

*For every ∊* > 0 *there exists K* ∈ *𝓚*(*X*) *such that ||T*(*f*)*||*_{F} ≤ *∊ if f* ∈ *C*_{b}(*X*, *E*) *with ||f||* ≤ 1 *and f|*_{K} ≡ 0.

$\tilde{m}$(*X*) *= ||T||*.

*Every f* ∈ *C*_{b}(*X*, *E*) *is m-integrable over X* in *the Riemann-Stieltjes sense and for y′* ∈ *F′*,
$$\left({\displaystyle \underset{X}{\int}f\text{\hspace{0.17em}}dm}\right)\left({y}^{\prime}\right)={\displaystyle \underset{X}{\int}f\text{\hspace{0.17em}}d{m}_{{y}^{\prime}}}.$$

*∫*_{X} h d m = (*I*)*∫*_{X} h d m for h ∈ *C*_{rc}(*X*, *E*).

*For f* ∈ *C*_{b}(*X*, *E*), *∫*_{X} f d m ∈ *i*_{F}(*F*) *and T*(*f*) *= j*_{F}(*∫*_{X} f d m).

*Conversely*, *assume that a measure m* ∈ *M*(*X*, *𝓛*(*E*, *F″*)) *satisfies the conditions* (iii) *and* (iv). *Then there exists a unique* (*β*, || · ||_{F})*-continuous operator T* : *C*_{b}(*X*, *E*) → *F such that* (ii), (v), (vi), (vii), (viii) *and* (ix) *hold and m coincides with the representing measure of T*.

*Proof*. For *y′* ∈ *F′*, we have $\left({y}^{\prime}\circ T\right)\left(f\right)={\displaystyle {\int}_{X}f{d}_{{\mu}_{{y}^{\prime}\circ T}}}$ for *f* ∈ *C*_{b}(*X*, *E*), where *μ*_{y}_{′◦T} ∈ *M*(*X*, *E*^{′}) (see Theorem 2.4). Then for *A* ∈ *𝓑o* and *x* ∈ *E* we have

$$\begin{array}{l}{m}_{{y}^{\prime}}\left(A\right)\left(x\right)=\left(m\left(A\right)\left(x\right)\right)\left({y}^{\prime}\right)=\widehat{T}\left({\mathrm{\U0001d7d9}}_{A}\otimes x\right)\left({y}^{\prime}\right)=\pi \left({\mathrm{\U0001d7d9}}_{A}\otimes x\right)\left({y}^{\prime}\circ T\right)\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\pi \left({\mathrm{\U0001d7d9}}_{A}\otimes x\right)\left({\text{\Phi}}_{{\mu}_{{y}^{\prime}\circ T}}\right)=\left(I\right){\displaystyle \underset{X}{\int}\left({\mathrm{\U0001d7d9}}_{A}\otimes x\right)d{\mu}_{{y}^{\prime}\circ T}={\mu}_{{y}^{\prime}\circ T}\left(A\right)\left(x\right),}\hfill \end{array}$$i.e., *m*_{y′} = μ_{y′◦T} ∈ *M*(*X*, *E′*). Since $\tilde{m}\left(X\right)=\Vert \widehat{T}\Vert <\infty $ (see [1, §9, Theorem 1]), the condition (i) holds. Moreover, *y′*(*T*(*f*)) *= ∫*_{X} f dm_{y′} for *f* ∈ *C*_{b}(*X*, *E*), i.e., (ii) holds. Using (i) and (ii) we easily obtain that (iii) holds. Since the family {*y′ ◦ T* : *y′* ∈ *B*_{F′}} is *β*-equicontinuous, by Corollary 2.7 and (ii) for every *∊* > 0 there exists *K* ∈ *𝓚*(*X*) such that $\tilde{m}\left(X\backslash K\right)={\mathrm{sup}}_{{y}^{\prime}\in {B}_{{F}^{\prime}}}\left|{m}_{{y}^{\prime}}\right|\left(X\backslash K\right)\le \epsilon $ and this means that (iv) holds. Then by (iv), (ii) and Lemma 2.5 we easily obtain that (v) holds. Using (ii) and Lemma 2.5, we have

$$\begin{array}{ll}\Vert T\Vert \hfill & =\mathrm{sup}\left\{\left|{\displaystyle \underset{X}{\int}f\text{\hspace{0.17em}}d{m}_{{y}^{\prime}}}\right|:f\in {C}_{b}\left(X,E\right),\Vert f\Vert \le \mathrm{\U0001d7d9},{y}^{\prime}\in {B}_{{F}^{\prime}}\right\}\hfill \\ \hfill & =\mathrm{sup}\left\{\left|{m}_{{y}^{\prime}}\right|\left(X\right):{y}^{\prime}\in {B}_{{F}^{\prime}}\right\}=\tilde{m}\left(X\right),\hfill \end{array}$$i.e., (vi) holds.

Now we shall show that every f ∈ *C*_{b}(*X*, *E*) is *m*-integrable over *X*. Indeed, let *f* ∈ *C*_{b}(*X*, *E*) and *η >* 0 be given, and $\epsilon =\frac{\eta}{4\tilde{m}\left(X\right)}$. By (iv) there exists *K* ∈ *𝓚*(*X*) such that ${\mathrm{sup}}_{{y}^{\prime}\in {B}_{{F}^{\prime}}}\left|{m}_{{y}^{\prime}}\right|\left(X\backslash K\right)\le \frac{\eta}{8\Vert f\Vert}$.

For each *t* ∈ *X* let *W*_{t} = {*s* ∈ *X* : *||f*(*s*) *– f*(*t*)*||*_{E} < ∊}. Hence there exists a set {*t*_{1}, …, *t*_{n}} in *X* such that $K\subset {\displaystyle \cup {}_{j=1}^{n}}\text{\hspace{0.17em}\hspace{0.17em}}{W}_{{t}_{i}}.$ Then $W:={\displaystyle \cup {}_{i=1}^{n}}\text{\hspace{0.17em}\hspace{0.17em}}{W}_{{t}_{i}}\in \text{\U0001d4d1}o$ and ${\mathrm{sup}}_{y\prime \in {B}_{F\prime}}\left|{m}_{y\prime}\right|\left(X\backslash W\right)\le \frac{\eta}{8\Vert f\Vert}.$ Define *A*_{1} = *W*_{t1}, ${A}_{i}={W}_{{t}_{i}}\backslash {\displaystyle \cup {}_{j=1}^{i-1}}\text{\hspace{0.17em}\hspace{0.17em}}{W}_{{t}_{j}}$ for *i =* 2, …, *n* and *A*_{n+}_{1} *= X \ W*. Then $W={\displaystyle \cup {}_{i=1}^{n}}{A}_{i}$ and *t*_{i} ∈ *A*_{i} for *i =* 1,..., *n* and let *t*_{n+}_{1} ∈ *A*_{n+}_{1}. Hence 𝓐_{∊} = {*A*_{1},..., *A*_{n}, *A*_{n+}_{1};*t*_{1},…*t*_{n}, t_{n}_{+1}} ∈ 𝓓_{X}. Let *𝓐*_{1}, *𝓐*_{2} ∈ 𝓓_{X} and 𝓐_{1} ≥ 𝓐_{∊}, 𝓐_{2} ≥ 𝓐_{∊}, where, 𝓐_{1} = {*B*_{1},..., *B*_{p},*s*_{1},...,*s*_{p}} and 𝓐_{2} = {*C*_{1},..., *C*_{q}, *τ*_{1},..., *τ*_{q}}. Let *J*_{i} = {*j* : *B*_{j} ⊂ *A*_{i}} for *i =* 1, …, *n+*1. Then arguing similarly as in the proof of Theorem 2.3 one can show that ${\Vert {S}_{{\text{\U0001d4d0}}_{\epsilon}}\left(f\right)-{S}_{{\text{\U0001d4d0}}_{i}}\left(f\right)\Vert}_{F\u2033}\le \frac{\eta}{2}$ for *i =* 1, 2, so ${\Vert {S}_{{\text{\U0001d4d0}}_{1}}\left(f\right)-{S}_{{\text{\U0001d4d0}}_{2}}\left(f\right)\Vert}_{F\u2033}\le \eta .$ This means that (*S*_{𝓐}(*f*)) is a Cauchy net in (*F*″, *||* · *||*_{F″}) and hence the integral *∫*_{X} f dm := lim_{𝓐} *S*_{𝓐}(*f*) exists in *F*″. This means that f is *m*-integrable over *X*.

Note that for each *y*′ ∈ * F'* and *𝓐 =* {*A*_{1},...,*A*_{n};*t*_{1},...,*t*_{n}) ∈ 𝓓_{X} we have ${S}_{\text{\U0001d4d0}}\left(f\right)\left(y\prime \right)={\displaystyle {\sum}_{i=1}^{n}{m}_{y\prime}\left({A}_{i}\right)\left(f\left({t}_{i}\right)\right)}$ and *∫*_{X} f dm_{y}_{′} = lim𝓐 *S*_{𝓐}(*f*)(*y*′). Hence we get,

$$\left({\displaystyle \underset{X}{\int}f\text{\hspace{0.17em}}dm}\right)\left({y}^{\prime}\right)=\underset{\text{\U0001d4d0}}{\mathrm{lim}}{S}_{\text{\U0001d4d0}}\left(f\right)\left({y}^{\prime}\right)={\displaystyle \underset{X}{\int}f\text{\hspace{0.17em}}d{m}_{{y}^{\prime}}}.$$(7)Thus (vii) holds. Now we shall show that the Integration operator *S*_{m} : *C*_{b}(*X*, *E*) → *F*″ defined by the equation

$$\begin{array}{l}\\ {S}_{m}\left(f\right):={\displaystyle \underset{X}{\int}f\text{\hspace{0.17em}}dm\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}f}\in {C}_{b}\left(X,E\right)\end{array}$$is (*β*, || · ||_{F″})-continuous. Indeed, let *∊* > 0 be given. Since {*y′ ◦ T* : *y*′ ∈ *B*_{F}_{′}} is *β*-equicontinuous and (ii) holds, there exists a neighbourhood *V* of 0 for *β* such that ${\mathrm{sup}}_{y\prime \in {B}_{F\prime}}\left|{\displaystyle {\int}_{X}\text{\hspace{0.17em}\hspace{0.17em}}f\text{\hspace{0.17em}\hspace{0.17em}}d{m}_{y\prime}}\right|\le \epsilon $ for *f* ∈ *V*. Assume that (*f*_{α}) is a net in *C*_{b}(*X*, *E*) such that *f*_{α} → 0 for *β*. Then one can choose *α*_{∊} such that *f*_{α} ∈ *V* for *α* ≥ *α*_{∊}. Hence for *α* ≥ *α*_{∊} we have,

$$\begin{array}{ll}{\Vert {S}_{m}\left({f}_{\alpha}\right)\Vert}_{{F}^{\u2033}}\hfill & =\text{sup}\left\{\left|\left({\displaystyle \underset{X}{\int}{f}_{\alpha}\text{\hspace{0.17em}}\text{\hspace{0.17em}}dm}\right)\left({y}^{\prime}\right)\right|:{y}^{\prime}\in {B}_{{F}^{\prime}}\right\}\hfill \\ \hfill & =\mathrm{sup}\left\{\left|{\displaystyle \underset{X}{\int}{f}_{\alpha}\text{\hspace{0.17em}}\text{\hspace{0.17em}}d{m}_{{y}^{\prime}}}\right|:{y}^{\prime}\in {B}_{{F}^{\prime}}\right\}\le \epsilon ,\hfill \end{array}$$i.e., *S*_{m} is (*β*, || · ||_{F″})-continuous. Using (ii), (7) and (6) for *h* ∈ *C*_{rc}(*X*, *E*) and *y′* ∈ *F*^{′}, we have

$${S}_{m}\left(h\right)\left({y}^{\prime}\right)=\left({\displaystyle \underset{X}{\int}h\text{\hspace{0.17em}}\text{\hspace{0.17em}}dm}\right)\left({y}^{\prime}\right)={\displaystyle \underset{X}{\int}h\text{\hspace{0.17em}}\text{\hspace{0.17em}}d{m}_{{y}^{\prime}}}={y}^{\prime}\left(T\left(h\right)\right)=\widehat{T}\left(h\right)\left({y}^{\prime}\right).$$It follows that ${S}_{m}\left(h\right)=\widehat{T}\left(h\right),$ where $\widehat{T}\left(h\right)\in {i}_{F}\left(F\right).$ Hence

$$\underset{X}{\int}h\text{\hspace{0.17em}}\text{\hspace{0.17em}}dm=\left(I\right){\displaystyle \underset{X}{\int}h\text{\hspace{0.17em}}\text{\hspace{0.17em}}dm\in {i}_{F}\left(F\right).}$$Thus the condition (viii) is satisfied. Assume now that f ∈ *C*_{b}(*X*, *E*) and choose a net (*h*_{α}) in *C*_{rc}(*X*, *E*) such that *h*_{α} → *f* for *β*. Then

$$\begin{array}{l}\\ {\displaystyle \underset{X}{\int}f\text{\hspace{0.17em}}dm}={S}_{m}\left(f\right)=\underset{\alpha}{\mathrm{lim}}{S}_{m}\left({h}_{\alpha}\right)=\underset{\alpha}{\mathrm{lim}}{\displaystyle \underset{X}{\int}{h}_{\alpha}dm,}\end{array}$$and hence *∫*_{X} f dm ∈ *i*_{F}(*F*) because *∫*_{X} h_{α}dm ∈ *i*_{F}(*F*). It follows that

$$T\left(f\right)=\underset{\alpha}{\mathrm{lim}}T\left({h}_{\alpha}\right)=\underset{\alpha}{\mathrm{lim}}{j}_{F}\left({\displaystyle \underset{X}{\int}{h}_{\alpha}\text{\hspace{0.17em}}\text{\hspace{0.17em}}dm}\right)={j}_{F}\left(\underset{\alpha}{\mathrm{lim}}{\displaystyle \underset{X}{\int}{h}_{\alpha}\text{\hspace{0.17em}}\text{\hspace{0.17em}}dm}\right)={j}_{F}\left({\displaystyle \underset{X}{\int}f\text{\hspace{0.17em}}\text{\hspace{0.17em}}dm}\right),$$i.e., (ix) holds.

Assume that *m* ∈ *M*(*X*, *𝓛*(*E*, *F*^{″})) satisfies the conditions (iii) and (iv). For *f* ∈ *C*_{b}(*X*, *E*) define a linear mapping Ψ_{f} on *F*^{′} by Ψ_{f}(*y*′) *= ∫*_{X} f dm_{y}_{′} for *y′* ∈ *F*^{′}. Then Ψ_{f} is *σ*(*F*^{′}, *F*)-continuous, so there exists a unique *y*_{f} ∈ *F* such that Ψ_{f} = *i*_{F}(*y*_{f}),*i.e*., Ψ_{f}(*y*′) *= y*^{′}(*y*_{f}) for *y′* ∈ *F*^{′}. Define a mapping *T* : *C*_{b}(*X*, *E*) → *F* by *T*(*f*) := *y*_{f}. Then for *y*′ ∈ *F*′, (*y*′ ◦ *T*)(*f*) = *y*′(*y*_{f}) *= ∫*_{X} f dm_{y}_{′} for *f* ∈ *C*_{b}(*X*, *E*), i.e., (ii) holds. Hence by Corollary 2.7 the set {*y*′ ◦ *T* : *y*′ ∈ *B*_{F}_{′}} is *β*-equicontinuous, and this means that *T* is (*β*, || · ||_{F})-continuous.

Let *m*_{o} : *𝓑o* → *𝓛*(*E*, *F"*) be the representing measure of *T*. Then by the first part of theorem *m*_{o} ∈ *M*(*X*, *𝓛*(*E*, *F"*)) and for each *y*′ ∈ *F*′, we have *y*′(*T*(*f*)) *= ∫*_{X} f d(*m*_{o})_{y′} for *f* ∈ *C*_{b}(X, *E*). It follows that (*m*_{o})_{y′} = *m*_{y′}, so *m*_{o} = *m*. In view of the first part of the theorem, the conditions (v), (vi), (vii), (viii) and (ix) are satisfied. □

Assume that *T* : *C*_{b}(*X*, *E*) → *F* is a (*β*, || ·||_{F})-continuous operator and *m* is its representing measure. For *x* ∈ *E* let us set,

$${T}_{x}\left(u\right):=T\left(u\otimes x\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}u\in {C}_{b}\left(X\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{m}_{x}\left(A\right):=m\left(A\right)\left(x\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}A\in \text{\U0001d4d1}o.$$Then *T*_{x} : *C*_{b}(*X*) → *F* is a (*β*, || · ||_{F})-continuous operator. Let $\chi :B\left(\text{\U0001d4d1}o\right)\to {C}_{b}{\left(X\right)}_{\beta}^{\u2033}$ stand for the canonical embedding, i.e., for *υ* ∈ *B*(*𝓑o*),

$$\chi \left(\upsilon \right)\left({\phi}_{v}\right):=\left(I\right){\displaystyle \underset{X}{\int}\upsilon \text{\hspace{0.17em}}dv\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}v\in M\left(X\right),$$where *φ*_{v}(*u*) = *∫*_{X}u dv for *u* ∈ *C*_{b}(*X*). Define

$${\widehat{T}}_{x}:={\left({T}_{x}\right)}^{\prime \text{}\prime}\circ \chi :B\left(\text{\U0001d4d1}o\right)\to {F}^{\u2033}.$$Then ${\widehat{T}}_{x}\left({C}_{b}\left(X\right)\right)\subset {i}_{F}\left(F\right)$ and ${T}_{x}\left(u\right)={j}_{F}\left({\widehat{T}}_{x}\left(u\right)\right)$ for *u* ∈ *C*_{b}(*X*).

*Let T* : *C*_{b}(*X*, *E*) → *F be a* (*β*, *||* · *||*_{F})*-continuous operator and m be its representing measure. Then for each x* ∈ *E the following statements hold*:

*T"*(*π*(𝟙_{A} ⊗ *x*)) = (*T*_{x})*"*(*χ*(𝟙_{A})) *for A* ∈*𝓑o*.

${m}_{x}\left(A\right)={\widehat{T}}_{x}\left({1}_{A}\right)$ *for A* ∈ *𝓑o*.

${\widehat{T}}_{x}\left(\upsilon \right)=\left(I\right){\displaystyle {\int}_{X}\text{\hspace{0.17em}\hspace{0.17em}}\upsilon d{m}_{x}}$ *for υ* ∈ *B*(*𝓑o*).

*Proof*. Let *y′* ∈ *F*^{′}. Then by Theorem 3.2 for *u* ∈ *C*_{b}(*X*), we have

$$\left({y}^{\prime}\circ {T}_{x}\right)\left(u\right)={y}^{\prime}\left(T\left(u\otimes x\right)\right)={\displaystyle \underset{X}{\int}\left(u\otimes x\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}d{m}_{{y}^{\prime}}={\displaystyle \underset{X}{\int}u\text{\hspace{0.17em}}\text{\hspace{0.17em}}d{m}_{x,{y}^{\prime}}}={\phi}_{{m}_{x,{y}^{\prime}}}\left(u\right),}$$where ${\phi}_{{m}_{x,y\prime}}\in {C}_{b}{\left(X\right)}_{\beta}^{\prime}.$ Hence, we get

$$\begin{array}{l}{\left({T}_{x}\right)}^{\prime \text{}\prime}\left(\chi \left({\mathrm{\U0001d7d9}}_{A}\right)\right)\left({y}^{\prime}\right)=\chi \left({\mathrm{\U0001d7d9}}_{A}\right)\left({\left({T}_{x}\right)}^{\prime}\left({y}^{\prime}\right)\right)=\chi \left({\mathrm{\U0001d7d9}}_{A}\right)\left({y}^{\prime}\circ {T}_{x}\right)\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\chi \left({\mathrm{\U0001d7d9}}_{A}\right)\left({\phi}_{{m}_{x,{y}^{\prime}}}\right)={m}_{x,{y}^{\prime}}\left(A\right)={m}_{x}\left(A\right)\left({y}^{\prime}\right).\hfill \end{array}$$On the other hand, by Theorem 3.2, $y\prime \text{\hspace{0.17em}\hspace{0.17em}}\circ \text{\hspace{0.17em}\hspace{0.17em}}T={\text{\Phi}}_{{m}_{y\prime}},$ and hence

$$\begin{array}{l}{T}^{\u2033}\left(\pi \left({\mathrm{\U0001d7d9}}_{A}\otimes x\right)\right)\left({y}^{\prime}\right)=\pi \left({\mathrm{\U0001d7d9}}_{A}\otimes x\right)\left({T}^{\prime}\left({y}^{\prime}\right)\right)=\pi \left({\mathrm{\U0001d7d9}}_{A}\otimes x\right)\left({y}^{\prime}\circ T\right)\hfill \\ =\pi \left({\mathrm{\U0001d7d9}}_{A}\otimes x\right)\left({\text{\Phi}}_{{m}_{{y}^{\prime}}}\right)=\left(I\right){\displaystyle \underset{X}{\int}\left({\mathrm{\U0001d7d9}}_{A}\otimes x\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}d{m}_{{y}^{\prime}}}={m}_{x}\left(A\right)\left({y}^{\prime}\right).\hfill \end{array}$$It follows that (i) holds. For *A* ∈ *𝓑o*, we have

$${m}_{x}\left(A\right):=\widehat{T}\left({\mathrm{\U0001d7d9}}_{A}\otimes x\right)={T}^{\u2033}\left(\pi \left({\mathrm{\U0001d7d9}}_{A}\otimes x\right)\right)={\left({T}_{x}\right)}^{\prime \text{}\prime}\left(\chi \left({\mathrm{\U0001d7d9}}_{A}\right)\right)={\widehat{T}}_{x}\left({\mathrm{\U0001d7d9}}_{A}\right),$$and hence, ${\widehat{T}}_{x}\left(\upsilon \right)=\left(I\right)\text{\hspace{0.17em}\hspace{0.17em}}{\displaystyle {\int}_{X}\text{\hspace{0.17em}\hspace{0.17em}}\upsilon \text{\hspace{0.17em}\hspace{0.17em}}d{m}_{x}}$ for *υ* ∈ *B*(*𝓑o*). Thus (ii) and (iii) are satisfied. □

Dobrakov [20, Theorem 2] showed that if *X* is a locally compact Hausdorff space and *T* : *C*_{0}(*X*, *E*) → *F* is a bounded linear operator, then its representing measure takes its values in *𝓛*(*E*, *F*) if and only if *T*_{x} : *C*_{0}(*X*) → *F* is weakly compact for each *x* ∈ *E*. Now we prove the analogue of this result for (*β*, || · ||_{F})-continuous operators *T* : *C*_{b}(*X*, *E*) → *F*, where *X* is a completely regular Hausdorff space (see [26, Theorem 4.4]).

*Let T* : *C*_{b}(*X*, *E*) → *F be a* (*β*, || · ||_{F})*-continuous operator and m be its representing measure. Then for each x* ∈ *E the following statements are equivalent*:

*T*_{x} : *C*_{b}(*X*) → *F is weakly compact*.

*m*(*A*)(*x*) ∈ *i*_{F}(*F*) *for each A* ∈ *𝓑o*.

*m*_{x} : *𝓑o* → *F*″ : *countably additive*.

*m*_{x} : *𝓑*o → *F*″ *is strongly bounded*.

*Proof*. (i) ⇒ (ii) Assume that *T*_{x} is weakly compact. in view of [42, Corollary 9.3.2] it follows that $\left({T}_{x}\right)\u2033\left({C}_{b}{\left(X\right)}_{\beta}^{\u2033}\right)\subset {i}_{F}\left(F\right).$ Hence ${\widehat{T}}_{x}\left(B\left(\text{\U0001d4d1}o\right)\right)\subset {i}_{F}\left(F\right)$ and by Lemma 3.3 *m*_{x}(*A*) ∈ *i*_{F}(*F*) for *A* ∈ 𝓑*o*.

(ii) ⇒ (iii) Assume that *m*(*A*)(*x*) ∈ *i*_{F}(*F*) for *A* ∈ *𝓑o*, and let (*m*_{F})_{x}(*A*) *:= j*_{F}(*m*(*A*)(*x*)) for *A* ∈ *𝓑o*. Then (*y*′ ◦ (*m*_{F})_{x})(*A*) = (*m*(*A*)(*x*))(*y*^{′}) = *m*_{x, y}_{′} (*A*) for *A* ∈ *𝓑o*. in view of Theorem 3.2 for each *y*′ ∈ *F′*, *m*_{y′} ∈ *M*(*X*, *E*′) and it follows that *m*_{x}_{,y′} is countably additive. Hence *y*^{′} *◦* (*m*_{F})_{x} is countably additive, and by the Orlicz-Pettis theorem (*m*_{F})_{x} : *𝓑o* → *F* is countably additive. Hence *m*_{x} : 𝓑*o* → *F*″ is also countably additive.

(iii) ⇒ (iv) It is obvious.

(iv) ⇒ (i) Assume that *m*_{x} : *𝓑o* → *F*″ is strongly bounded. Then in view of Lemma 3.3 and [45, Theorem 1,

p. 148] $\widehat{T}$_{x} : *B*(*𝓑o*) → *F*″ is weakly compact. Since *T*_{x}(*u*) *= j*_{F}($\widehat{T}$_{x}(*u*)) for *u* ∈ *C*_{b}(*X*), we obtain that *T*_{x} is weakly compact. □

Assume that *T* : *C*_{b}(*X*, *E*) → *F* is a (*β*, *||* · *||*_{F})-continuous operator such that *T*_{x} : *C*_{b}(*X*) → *F* is weakly compact for each *x* ∈ *E*. From now on we will write:

$${m}_{F}\left(A\right)\left(x\right):={j}_{F}\left(m\left(A\right)\left(x\right)\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}A\in \text{\U0001d4d1}o,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in E.$$Then *m*_{F} : 𝓑o → 𝓛(*E*, *F*) and $\tilde{{m}_{F}}\left(A\right)=\tilde{m}\left(A\right)$ for *A* ∈ 𝓑*o*.

By 𝓣_{s} we denote the topology of simple convergence in *ca*(*𝓑o*). Then *𝓣*_{s} is generated by the family {*P*_{A} : *A* ∈ *𝓑o*} of seminorms, where *P*_{A}(*v*) *= |v*(*A*)*|* for *v* ∈ *ca*(*𝓑o*). Note that *M(X*) is a closed subspace of the Banach space *ca*(*𝓑o*) (see Remark 2.6).

A completely regular Hausdorff space *X* is said to be a *k-space* if each set which meets every compact subset in a closed set must be closed. *X* is a *k*-space, for instance if *X* is locally compact or first countable (see [38, Chap. 3, § 3], [6, p. 107]). Making use of [43] we can state the following extension to *k*-spaces of the celebrated Dieudonné-Grothendieck’s criterion on relative weak compactness in the space *M*(*X*) (see [44, Theorem 2], [45, Theorem 14, p. 98-103]), which will play a crucial role in the study of operators on *Cb*(*X*, *E*).

*Assume that X is a k-space and 𝓜 is a subset of M*(*X*) *such that* sup*v*∈𝓜 |*v*|(*X*) < ∞. *Then the following statements are equivalent*:

*𝓜 is relatively weakly compact* in *the Banach space M*(*X*).

*𝓜 is uniformly countably additive*, *i.e*., sup_{v∈𝓜} |*v*(*A*_{n})| → 0 *whenever A*_{n} ↓ ∅, (*A*_{n}) ⊂ *𝓑o*.

*𝓜 is uniformly strongly additive*, *i.e*., sup_{v∈𝓜} |*v*(*A*_{n})| → 0 *whenever* (*A*_{n}) *is pairwise disjoint in 𝓑o*.

sup_{v∈𝓜} |*v|*(*A*_{n}) → 0 *whenever* (*A*_{n}) *is a pairwise disjoint sequence in 𝓑o*.

*𝓜 is a relatively 𝓣*_{s}-compact subset of M(*X*).

*𝓜 is uniformly regular*, *i.e*., *for every A* ∈ *𝓑o and ∊* > 0 *there exist K* ∈ *𝓚*(*X*) *and O* ∈ 𝓣 *with K* ⊂ *A* ⊂ *O such that* sup_{v∈𝓜} |*v*|(*O \ K*) ≤ *∊*.

sup_{v∈𝓜} |*v*|(*O*_{n}) → 0 *whenever* (*O*_{n}) *is a pairwise disjoint sequence in* 𝓣.

*Proof*. (i) ⇔ (ii) It follows from [46, Chap. 7, Theorem 13] because *M*(*X*) is a closed subset of the Banach space *ca*(*𝓑o*) (see [47, Chap. 3, §3, Corollary 3]).

(ii) ⇔(iii) See [46, Theorem 10, pp. 88-89].

(iii) ⇔(iv) See [45, Proposition 17, p. 8].

(iv) ⇔ (v) ⇔ (vi) It follows [43, Remark, pp. 211-212]. (iv) ⇔ (vii) It is obvious.

(vii) ⇔ (iv) Assume that (vii) holds. Then by [43, Theorem 8] the set {|*v*| : *v* ∈ *𝓜*} is relatively 𝓣_{s}-compact In *M*(*X*). Hence by [43, Remark, pp. 211-212] the condition (iv) holds. □

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