UDC 517.397:519.53:519.41/47:517.9

MATHEMATICS

V. I. RYBAKOV

THE RADON–NIKODYM THEOREM AND THE REPRESENTATION OF VECTOR MEASURES BY AN INTEGRAL

(Presented by Academician P. S. Novikov on 7 VII 1967)

The paper considers a generalization of the Radon–Nikodym theorem to the case where both measures are vector-valued (with values in a Banach space). A similar situation was considered in paper (¹), but in (¹) rather severe requirements were imposed on vector measures. In the present paper some of these restrictions are removed; however, it is necessary to introduce a more general integral in order to represent one vector measure with respect to another (in (¹) the bilinear Bartle integral is used for this purpose). As one of the applications of such a representation, a description will be given of vector measures of σ-finite variation with values in a reflexive space. In conclusion, necessary and sufficient conditions will be given for representing a vector measure by a Bochner integral.

By \(\Sigma\) we shall denote a certain σ-algebra of subsets of a set \(S\). We shall call a countably additive function \(m\), defined on \(\Sigma\) and taking values in a Banach space, a vector measure, i.e., \(m\) is such that for any sequence of pairwise disjoint sets

\[ E_i \in \Sigma \quad\text{one has}\quad m\left(\bigcup_{i=1}^{\infty} E_i\right)=\sum_{i=1}^{\infty} m(E_i), \]

where the series on the right-hand side converges unconditionally.

By the variation of the vector measure \(m\) on the set \(E \in \Sigma\) (notation \(v(m,E)\)) we mean

\[ \sup \left\{ \sum_i \|m(E_i)\| \right\}, \]

where the supremum is taken over all finite and countable systems \(\{E_i\}\) of pairwise disjoint sets \(E_i \subset E\), \(E_i \in \Sigma\).

If \(v(m,S)<\infty\), then \(m\) is said to have finite variation. If

\[ S=\bigcup_{n=1}^{\infty} E_n, \]

where \(E_n \in \Sigma\) are such that \(v(m,E_n)<\infty\), \(n=1,2,\ldots\), then we shall say that \(m\) has σ-finite variation.

It is said that a vector measure \(m_2:\Sigma \to Y\) is absolutely continuous with respect to \(m_1:\Sigma \to X\) \((m_2 \ll m_1)\), if for \(E \in \Sigma\) from \(v(m_1,E)=0\) it follows that \(m_2(E)=0\).

Theorem 1. In order that a vector measure \(m\), absolutely continuous with respect to a positive measure \(\mu\), have σ-finite variation, it is necessary and sufficient that the set \(S\) can be represented in the form

\[ S=\bigcup_{N=1}^{\infty} E_N, \]

where \(E_N \in \Sigma\) are such that if \(F \subset E_N\), \(F \in \Sigma\), then

\[ \|m(F)\| \le N\mu(F). \]

By \(\pi\) we shall denote any finite set of natural numbers; \(\pi \ge \pi_1\) means that \(\pi \supset \pi_1\). Let \((Y,\tau)\) be a separable locally convex space and let \(V\) be some neighborhood of zero in the topology \(\tau\).

Definition. A series \(\sum_i y_i\) \((y_i \in Y,\ i=1,2,\ldots)\) will be called unconditionally summable with accuracy up to \(V\) to \(y_0 \in Y\) if there exists a \(\pi_0\) such that for \(\pi \le \pi_0\)

\[ y_0-\sum_{i\in\pi} y_i \in V. \]

Consider a certain bilinear operator \(u: Z \times X \to Y\), where \(X,Y,Z\) are arbitrary spaces. For brevity, instead of \(u(z,x)\) we shall write \(zx\). In what follows it is assumed that \(X\) is a Banach space, and that \(Y\) is endowed with a certain separable locally convex topology \(\tau\).

By a partition we shall mean any finite or countable family
\(\Delta=\{E_i\}\), \(i=1,2,\ldots\), of pairwise disjoint sets from \(\Sigma\) such that \(\bigcup_i E_i=S\). If \(\Delta'=\{F_j\}\), then \(\Delta'\geq \Delta\) means that each set \(F_j\in\Delta'\) is a subset of some \(E_i\in\Delta\).

Definition. A function \(f:S\to Z\) is called \(\tau\)-integrable on \(S\) (with respect to \(m:\Sigma\to X\)), and its \(\tau\)-integral on \(S\) is \(y_S\in Y\), if for an arbitrary neighborhood of zero \(V\) in \(\tau\) there is a partition \(\Delta_V\) such that, for \(\Delta\geq \Delta_V\), every series
\(\sum_i f(s_i)m(E_i)\) \((s_i\in E_i,\ E_i\in\Delta)\) is unconditionally summable with accuracy up to \(V\) to \(y_S\).

Thus, by definition,
\[ y_S=(\tau)\int_S f(s)\,dm=(\tau)\int_S f\,dm \]
(if the function \(f\) is \(\tau\)-integrable on \(S\), then the \(\tau\)-integral is determined uniquely). It should be noted that the \(\tau\)-integral depends, of course, on the bilinear operator \(u:Z\times X\to Y\), and one ought to speak of the \((\tau,u)\)-integral with respect to \(m\); but for simplicity we shall adhere to the terminology introduced above.

Definition. A function \(f:S\to Z\) is called \(\tau\)-integrable if, for every \(E\in\Sigma\), the function \(f(s)\chi_E(s)\) is \(\tau\)-integrable on \(S\) (\(\chi_E\) is the characteristic function of the set \(E\)). By definition,
\[ (\tau)\int_E f\,dm=(\tau)\int_S f(s)\chi_E(s)\,dm. \]

If \(m\) is a positive measure, \(Z=Y\), and \(u(y,a)=ay\) \((a\in X,\ y\in Y)\), then our \(\tau\)-integral coincides with Phillips’ \(U\)-integral, see \((^2)\).

The following assertion is easily proved: if the functions \(f\) and \(g\) are \(\tau\)-integrable, then the function \(\alpha f+\beta g\) is \(\tau\)-integrable (\(\alpha,\beta\) are scalars), and for \(E\in\Sigma\)
\[ (\tau)\int_E(\alpha f+\beta g)\,dm = \alpha\cdot(\tau)\int_E f\,dm + \beta\cdot(\tau)\int_E f\,dm. \]

Lemma. Let \(f:S\to Z\) be \(\tau\)-integrable. If \(\Delta=\{E_i\}\) is a partition of \(S\) such that, for \(\Delta'=\{G_j\}\geq\Delta\), every series of the form \(\sum_j f(s_j)m(G_j)\) \((s_j\in G_j)\) is unconditionally summable with accuracy up to \(V\) to \((\tau)\int f\,dm\), then for any \(E\in\Sigma\) the series \(\sum_i f(s_i)m(E_i\cap E)\) \((s_i\in E_i\cap E,\ E_i\in\Delta)\) is unconditionally summable with accuracy up to \(3V\) to \((\tau)\int_E f\,dm\).

Theorem 2 (on countable additivity of the \(\tau\)-integral). If the function \(f:S\to Z\) is \(\tau\)-integrable, then for any sequence of pairwise disjoint sets \(E_i\in\Sigma\) and any neighborhood of zero \(V\) in \(\tau\) there exists a \(\pi_V\) such that for every \(\pi\geq\pi_V\) one has
\[ (\tau)\int_{\bigcup_{i=1}^{\infty}E_i} f\,dm - \sum_{i\in\pi}(\tau)\int_{E_i} f\,dm \in V. \]

Using the integral introduced, we shall give a generalization of the Radon–Nikodym theorem.

Let \(X^*,Y^*\) be the spaces dual respectively to the Banach spaces \(X,Y\). Define the bilinear operator \(u:L(X^*,Y^*)\times X^*\to Y^*\) as follows*:
\[ u(z,x^*)=z(x^*),\quad z\in L(X^*,Y^*),\ x^*\in X^*. \]
If, further, \((Y^*,\tau)\) is the space \(Y^*\) in its \(Y\)-topology (see \((^3)\), p. 453, definition 2), then the following holds

* By \(L(X^*,Y^*)\) is denoted, as usual, the space of bounded linear operators from \(X^*\) into \(Y^*\).

Theorem 3. Let \(m_1:\Sigma\to X^*\), \(m_2:\Sigma\to Y^*\) be vector measures, and let \(\mu_1\) be the variation of \(m_1\). Suppose, further, that: 1) \(\mu_1\) is a full* finite measure on \(\Sigma\); 2) the measure \(m_2\) has \(\sigma\)-finite variation; 3) \(m_2\ll m_1\).

Then there exists on \(S\) a function \(f\) with values in \(L(X^*,Y^*)\) such that

\[ m_2(E)=(\tau)\int_E f\,dm_1,\quad E\in\Sigma . \]

Theorem 4. Let \(m_1:\Sigma\to X\), \(m_2:\Sigma\to Y\) (\(X,Y\) are Banach spaces) be vector measures for which the following conditions are satisfied: 1) \(\nu(m_1,(\cdot))\) is a full finite measure on \(\Sigma\); 2) the variation \(m_2\) is \(\sigma\)-finite; 3) \(m_2\ll m_1\).

Then there exists on \(S\) a function \(f\) with values in \(L(X,Y^{**})\) such that

\[ m_2(E)=(\tau)\int_E f\,dm_1 . \]

In this assertion the bilinear operator \(u:L(X,Y^{**})\times X\to Y^{**}\) is defined by the equality \(u(z,x)=z(x)\), \(x\in X\), \(z\in L(X,Y^{**})\); for \(\tau\) on \(Y^{**}\) the \(Y^*\)-topology is taken, and \(y\) is the image of the element \(y\in Y\) under the natural embedding of \(Y\) into \(Y^{**}\).

It should be noted that under the conditions of Theorem 4: a) the function \(f\), generally speaking, is not uniquely determined, even up to a function equal to zero almost everywhere; b) it may happen that \(f(s)\notin L(X,Y)\) for almost all \(s\in S\). This, for example, will be the case when \(Y=L_1[0,1]\), \(S=[0,1]\), \(\Sigma\) is the collection of Lebesgue-measurable subsets of the interval \([0,1]\), \(m_2(E)=\chi_E\) for \(E\in\Sigma\), and \(m_1\) is Lebesgue measure on \(\Sigma\).

We give conditions sufficient for \(f(s)\in L(X,Y)\) for all \(s\in S\) in the case when \(m_1\) is a scalar measure (for other conditions, also in the case when \(m_1\) is a scalar measure, see (4), p. 269, Theorem 5).

Theorem 5. If, under the conditions of Theorem 4, \(m_1\) is a positive measure and the required function \(f\) is almost separably valued, then there exists a separably valued Pettis-integrable** function \(g:S\to Y\) such that

\[ m_2(E)=P\int_E g(s)\,dm_1 \]

for all \(E\in\Sigma\).

Let a positive full finite measure \(\mu\) be given on \(\Sigma\).

Theorem 6. If \(X\) is a reflexive space and the vector measure \(m\) (\(m:\Sigma\to X\)) is such that: 1) \(m\ll\mu\); 2) the variation \(m\) is \(\sigma\)-finite, then there exists a separably valued Pettis-integrable function \(f:S\to X\) such that

\[ m(E)=(P)\int_E f\,d\mu . \]

Corollary 1. In order that a vector measure \(m\), defined on \(\Sigma\) with values in a reflexive space \(X\), absolutely continuous with respect to \(\mu\), have \(\sigma\)-finite variation, it is necessary and sufficient that there exist a function \(f:S\to X\), Pettis-integrable, such that

\[ m(E)=(P)\int_E f\,d\mu . \]

Corollary 2. If \(X\) is reflexive, then for every Pettis-integrable function \(f:S\to X\) there exists a separably valued Pettis-integrable function \(g:S\to X\) such that

\[ (P)\int_E f\,dm=(P)\int_E g\,dm . \]

We give a theorem which is a generalization to the vector case of the result of G. P. Tolstov (see (6)). Let \((S,\Sigma,\mu)\) be a space with a full finite positive measure. We shall call a vector measure \(\lambda:\Sigma\to X\) (\(X\) a Banach space) elementary (with respect to \(\mu\)) if there exist \(F\in\Sigma\) and \(x\in X\) such that \(\lambda(E)=x\mu(E\cap F)\) for every \(E\in\Sigma\).

* A nonnegative measure \(\mu\), defined on \(\Sigma\), is called full if from \(E\in\Sigma\), \(F\subset E\), and \(\mu(E)=0\) it follows that \(F\in\Sigma\).

** For the definition of Pettis and Bochner integrals see, for example, (5), Ch. III.

Theorem 6. In order that a vector measure \(m:\Sigma \to X\) be representable as a Bochner integral of some function with respect to the measure \(\mu\), it is necessary and sufficient that there exist a sequence of elementary (with respect to \(\mu\)) measures \(\{\lambda_n\}\) \((\lambda_n:\Sigma \to X,\ n=1,2,\ldots)\) such that

\[ m(E)=\sum_{n=1}^{\infty}\lambda_n(E), \qquad \sum_{n=1}^{\infty} v(\lambda_n,S)<\infty . \]

Moscow State
Pedagogical Institute
named after V. I. Lenin

Received
23 VI 1967

REFERENCES

\({}^{1}\) M. Rao, Proc. Nat. Acad. Sci. U. S. A., 15, 771 (1964).
\({}^{2}\) R. Phillips, Trans. Am. Math. Soc., 47, 114 (1940).
\({}^{3}\) H. Dunford, J. Schwartz, Linear Operators, Moscow, 1962.
\({}^{4}\) N. Dinculeanu, Vector Measures, Berlin, 1966.
\({}^{5}\) E. Hille, R. Phillips, Functional Analysis and Semigroups, Moscow, 1962.
\({}^{6}\) G. P. Tolstov, Matem. sborn., 71 (113), 420 (1966).