Convergence and Differentiation
The reader is probably familiar with the extensive and penetrating theory of finite-dimensional linear spaces and operators thereon. One cannot hope to obtain a similarly penetrating theory when the vector space is infinite-dimensional without additional structure in the space. A first step in civilizing a linear space for the purposes of analysis is the introduction of a convenient topology, i.e. one with respect to which relevant operations are continuous. A natural candidate for such a topology in the linear space M of all complex-valued locally measurable functions is that of sequential pointwise convergence. However, this topology may fail to have cogent structural properties. One is led rather to deal with a modification \(\tilde M\) of M which is obtained by identifying functions which agree except on local null sets; and which is given the topology it inherits naturally from that of pointwise sequential convergence in M. It will be seen in fact that \(\tilde M\) is metrizable in this topology, in the key case of a finite measure space; and that it becomes a topological linear space in the following sense.
KeywordsMeasurable Function Linear Space Measure Space Function Class Bounded Variation
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