Chapter 1: Probability and Measure

\sigma-algebra

probability measure

measurable function

random variable/random vector

probability law/probability distribution

cumulative distribution function

probability density function

random variables X and Y are identically distributed if...

sub-\sigma-algebra

independent (\sigma-algebra)

independence (random variables)

Lebesgue integration
Step 1: Indicator functions

Lebesgue Integration
Step 2: simple functions

Lebesgue Integration
Step 3: non-negative random variables

Lebesgue Integration
Step 4: arbitrary random variables

integrable

null set

Properties of the integral
(five)

Properties of the Integral/Expectation
1. Linearity

Properties of the Integral/Expectation
2. Domination

Properties of the Integral/Expectation

3. the triangle inequality

Properties of the Integral/Expectation

4. Set bounded integrability

Properties of the Integral/Expectation

5. Change of variable

Dominated Convergence Theorem

Monotone Convergence Theorem

nth moment

mean
variance

moment generating function

Almost sure convergence

mean square convergence

Chebychev's Inequality

Cauchy-Schwarz inequality

convex function

Jensen's inequality

Absolutely Continuous measure

Equivalent measures

Radon-Nikodym

Radon-Nikodym derivative

stochastic process

integrable stochastic process