⑥ Measure theory
§1. \(\sigma\)-algebras¶
Algebras and \(\sigma\)-algebras¶
\( (\Omega, \mathcal{F}) \)
\( (\Omega, \mathcal{F}) \)
\(\sigma(\mathcal{C})\)
Borel \(\sigma\)-algebra¶
\(\mathcal{B^k}\)
§2. Measures¶
Measures¶
\( (\Omega, \mathcal{F}, \mu) \)
Probability measures¶
\( (\Omega, \mathcal{F}, P) \)
§3. Lebesgue integration¶
Lebesgue integration¶
Limits and Lebesgue integration¶
§4. \(L^p\) spaces¶
\(L^p\) spaces¶
§5. Random variables¶
Random variables¶
\(X(\cdot)\)
Distribution functions¶
\( F_{X}(\cdot) \)
§6. Mathematical expectations¶
Mathematical expectations¶
Some inequalties¶
Moment generating functions¶
§7. Conditional expectations¶
Conditional expectations¶
Law of iterated expectations¶
§8. Asymptotics¶
Convergence in probability¶
Convergence in distribution¶
The delta method¶
Law of large numbers¶
Central limit theorems¶
References¶
TBD