On some properties of the function space $\Cal M$
Commun. Korean Math. Soc. 2003 Vol. 18, No. 4, 677-685
Printed December 1, 2003
Joung Nam Lee
Seoul National University of Technology
Abstract : Let $M$ be the vector space of all real $S$-measurable functions defined on a measure space $(X, \Cal S, \mu )$. In this paper, we investigate some topological structure of $\Cal T$ on $\Cal M$. Indeed, $(M, \Cal T)$ becomes a topological vector space. Moreover, if $\mu$ is $\sigma$-finite, we can define a complete invariant metric on $\Cal M$ which is compatible with the topology $\Cal T$ on $\Cal M$, and hence $(M, \Cal T)$ becomes a $F$-space.
Keywords : $\mu$-equivalent, $\sigma$-finite measure, $S$-measurable function, $F$-space
MSC numbers : 28A33, 46E30
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