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0. Foundations

Summary

In this chapter we review several mathematical topics that form the foundation of probability and mathematical statistics. These include the algebra of sets and functions, general relations with special emphasis on equivalence relations and partial orders, counting measure, and some basic combinatorial structures such as permuations and combinations. We also discuss some more advanced topics from topology, measure theory and integration. Entire books have been written about the topics in the each of the sections in this chapter, so needless to say, the sections are brief summaries. You may wish to review the topics in this chapter as the need arises.

Sets, Functions, and Relations

  1. Sets
  2. Functions
  3. Relations
  4. Partial Orders
  5. Equivalence Relations

Cardinality and Combinatorics

  1. Cardinality
  2. Counting Measure
  3. Combinatorial Structures

Topology

  1. Topological Spaces
  2. Metric Spaces

Measure and Integration

  1. Measurable Spaces
  2. Measure Spaces
  3. Special Set Structures
  4. Existence and Uniqueness
  5. The Integral With Respect to a Measure
  6. Properties of the Integral
  7. General Measures
  8. Absolute Continuity and Density Functions
  9. Function Spaces

Apps

Sources and Resources

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