1. History of analysis

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