The Field Axioms of ℝ Notes

1.1. Modern analysis

1. Set Theory and Pathological Examples

• Cantor developed set theory and introduced the concept of cardinality and uncountability (e.g., ℝ is uncountable).

• Weierstrass and others constructed nowhere-differentiable continuous functions, showing that intuition could fail without rigor.

2. Measure Theory and Lebesgue Integration

• Lebesgue introduced a new way to define integration, based on measurable sets and functions, solving many of Riemann’s shortcomings.

• This became the foundation of modern probability theory and functional analysis.

3. Real Analysis Today

• Real analysis now includes:

  • Topology (open sets, compactness, connectedness)

  • Metric and normed spaces

  • Measure and integration theory

  • Applications in PDEs, ergodic theory, stochastic processes, and modern physics