Positivity 1: 255–270, 1997.
1997 Kluwer Academic Publishers. Printed in the Netherlands.
Image Measures and the So-Called Image Measure
at des Saarlandes, Fachbereich Mathematik, D-66041 Saarbr
(Received: 14 January 1997; accepted: 21 July 1997)
Abstract. The paper develops the formation of image measures on the basis of the recent monograph
of the author 1997. The main theorem says that the structure of so-called inner extensions carries
over from the initial measure to the image measure. One discloses the image measure catastrophe in
the sense of Laurent Schwartz 1973 to be a lack of inner regularity on the part of the initial measure.
Mathematics Subject Classiﬁcations (1991): 28A12, 28C15.
Keywords: image measure, image measure catastrophe, inner extensions of inner premeasures, Lusin
The introduction of the famous book  of Laurent Schwartz starts with a list of
three basic defects in traditional abstract measure theory, which in his words could
not have been mastered with adequate cohesion so far. The defects are
what he calls the catastrophe of the image measures;
the abstract product measure of two Borel-Radon measures can be a proper
restriction of their Borel-Radon product measure;
an abstract measure even on a Borel
algebra need not have a support.
The author notes that the theory of Radon measures on locally compact Hausdorff
topological spaces after Bourbaki does not have these defects. His crucial point is
that the same persists in the theory of Radon measures on arbitrary (for the most
part Hausdorff) topological spaces, which he sets out to develop in the ﬁrst part of
his book. It can thus be said that this theory combines the favourable aspects of the
two former ones.
Now the present author came to restructure the abstract theory of measure and
integration in the relevant fundamentals  (henceforth cited as MI). This work
clariﬁes the concepts and fortiﬁes the results of the abstract theory to quite some
extent. In particular it applies to the second and third of the above three defects, in
that it provides the proper contexts and then makes the defects disappear. This has
been done in MI chapter VII and section 9.
As to the ﬁrst of the three defects, the concept of image measures has not been
studied in MI. We do this in the present paper in the spirit of MI.