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Measure and Probability Theory

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Introduction to Uncertainty Quantification

Part of the book series: Texts in Applied Mathematics ((TAM,volume 63))

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Abstract

Probability theory, grounded in Kolmogorov’s axioms and the general foundations of measure theory, is an essential tool in the quantitative mathematical treatment of uncertainty. Of course, probability is not the only framework for the discussion of uncertainty: there is also the paradigm of interval analysis, and intermediate paradigms such as Dempster–Shafer theory, as discussed in Section 2.8 and Chapter 5.

To be conscious that you are ignorant is a great step to knowledge.

Sybil

Benjamin Disraeli

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Notes

  1. 1.

    It is an entertaining exercise to see what pathological properties can hold for a probability measures on a \(\sigma\)-algebra other than the power set of a finite set \(\mathcal{X}\).

  2. 2.

    Here we are again abusing notation to conflate \(C_{ij}: \mathcal{H}_{i} \oplus \mathcal{H}_{j} \rightarrow \mathbb{K}\) defined in (2.6) with \(C_{ij}: \mathcal{H}_{j} \rightarrow \mathcal{H}_{i}\) given by \(\langle C_{ij}(k_{j}),k_{i}\rangle _{\mathcal{H}_{i}} = C_{ij}(k_{i},k_{j})\).

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  1. Mathematics Institute, University of Warwick, Coventry, UK

    T. J. Sullivan

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  1. T. J. Sullivan
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Sullivan, T.J. (2015). Measure and Probability Theory. In: Introduction to Uncertainty Quantification. Texts in Applied Mathematics, vol 63. Springer, Cham. https://doi.org/10.1007/978-3-319-23395-6_2

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