Abstract
In the first three sections of this chapter we treat in detail the integration theory for functions on a compact topological space K. In the last section the results obtained will be applied to the proof of the Riesz–Markov–Kakutani theorem on the general form of linear functionals on the space C(K).
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Notes
- 1.
For the readers familiar with the theory of ordinal numbers (ordinals): Denote by \(\omega _1\) the first uncountable ordinal. Consider the set X of all ordinals that are not larger than \(\omega _1\). We call a neighborhood of the ordinal \(\alpha \) any subset \(U\subset X\) that contains an interval of the form \((\beta , \alpha ]\) with \(\beta <\alpha \). With the topology thus defined, the space X will be compact. Further, define a Borel measure \(\mu \) on X as follows: if the Borel set A contains a subset of the form \(B\setminus \{\omega _1\}\), where B is a closed set that has \(\omega _1\) as a limit point, then put \(\mu (A)=1\); otherwise, put \(\mu (A)=0\). Then we have \(\mu ((1,\omega _1))=1\), while at the same time the measure of any closed subset of the interval \((1, \omega _1)\) is equal to zero. Hence, \(\mu \) provides an example of a non-regular Borel measure on a compact topological space.
- 2.
The symbol \(\mathbbm {1}\) stands for the function identically equal to 1.
- 3.
This property justifies using the term “upper integral” also for the quantity \(\mathcal {I}^*\).
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Kadets, V. (2018). The Integral on C(K). In: A Course in Functional Analysis and Measure Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-92004-7_8
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DOI: https://doi.org/10.1007/978-3-319-92004-7_8
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