ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. 60, 2 (1991)
pp. 219-224

HILBERT-SPACE-VALUED MEASURES ON BOOLEAN ALGEBRAS (EXTENSIONS)
J. HAMHALTER and P. PTAK

Abstract. We prove that if $B_1$ is a Boolean subalgebra of $B_2$ and if $m\: B_1\to H$ is a bounded finitely additive measure, where $H$ is a Hilbert space, then $m$ admits an extension over $B_2$. This result generalizes the well-known result for real-valued measures (see e.g. Ref. 1). Then we consider orthogonal measures as a generalization of two-valued measures. We show that the latter result remains valid for $\dim H<\infty$. If $\dim H=\infty$, we are only able to prove a weaker result: If $B_1$ is a Boolean subalgebra of $B_2$ and $m\: B_1 \to H$ is an orthogonal measure, then we can find a Hilbert space $K$ such that $H\subset K$ and such that there is an orthogonal measure $k\: B_2\to K$ with $k/B_1=m$.

AMS subject classification. 06E99, 28B05
Keywords.

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