List of all definitions, results and examples

Below you can find a list of all definitions (blue/purple background, as elsewhere in the notes), key results i.e. theorems, lemmas, propositions etc. (red background, as elsewhere in the notes) and examples (green background, as elsewhere in the notes) that we have encountered throughout these notes, as a hopefully convenient overview! Recall that also in this list (as well as elsewhere in the notes), if you hoover over a link you will see a popup displaying the target of the link. Recall that an entry containing the symbol “⛔” is non-examinable as mentioned here.

Chapter 1:

Example 1.1: Some examples of implications and equivalences
Definition 1.2: Proof by contraposition and by contradiction
Example 1.3: Examples of proof by contraposition and by contradiction
Definition 1.6: Standard sets: empty, integers, rational numbers and real numbers
Definition 1.7: Sets on the real line: intervals
Definition 1.8: Relationships between sets
Definition 1.10: Operations on sets
Example 1.11: Infinite unions and intersections
Definition 1.12: Cartesian products
Definition 1.13: Open and closed subsets of the real line
Example 1.14: Open and closed subsets of the real line
Definition 1.16: Countable vs uncountable sets
Example 1.17: Examples of & rules for countable vs uncountable sets
Definition 1.20: Mutually disjoint
Definition 1.23: Partition
Example 1.24: Some examples of partitions
Definition 1.25: Collection of subsets
Example 1.26: Some collections of subsets
Definition 1.27: The powerset
Example 1.28: Simple examples of powersets
Example 1.29: Applying set relationships and operations to collections of subsets
Definition 1.30: A (probabilistic) experiment
Definition 1.31: Simple vs advanced experiment (ad hoc terminology!)
Definition 1.33: Sigma-algebras and measurable spaces
Example 1.34: Simple examples of sigma-algebras
Proposition 1.35: Some properties of sigma-algebras
Definition 1.36: Generating sigma-algebras
Example 1.37: Simple example of generating a sigma-algebra
Proposition 1.38: Characterisation of finite sigma-algebras
Example 1.40: Some examples of finite sigma-algebras
Definition 1.41: The Borel sigma-algebra
Proposition 1.45: The "level of information" a sigma-algebra represents
Example 1.47: Some examples of sigma-algebras and the "level of information" that they represent
Example 1.48: Finite sigma-algebras and the "level of information" that they represent
Definition 1.50: (Probability) measures and measure/probability spaces
Theorem 1.51: Properties of (probability) measures
Example 1.52: An example with (infinitely many) probability measures
Example 1.53: Probability measures vs empirical probabilities
Example 1.54: Probability measures on simple vs advanced experiments
Definition 1.55: Null events and almost sure events
Proposition 1.56: The Lebesgue measure and some of its properties
Example 1.58: Using the Lebesgue measure in a probability context
Lemma 1.59: An alternative generator of the Borel sigma-algebra [⛔]
Lemma 1.60: A characterisation of all finite measures on the Borel sigma-algebra [⛔]

Chapter 2:

Definition 2.1: A (probabilistic) experiment with one or more random variable(s)
Example 2.2: Example of random variables, and the technical condition required
Definition 2.3: Domain, co-domain, range and inverse images of a function
Example 2.4: Some examples of inverse images
Lemma 2.5: Some properties of inverse images [⛔]
Definition 2.6: Measurable functions
Example 2.8: Simple example of a measurable function
Proposition 2.9: About the sigma-algebra generated by a function
Example 2.10: The sigma-algebra generated by a constant function
Proposition 2.11: Alternative/sufficient conditions for measurability of functions
Example 2.12: Checking measurability of the constant function using the alternative strategy
Definition 2.13: Random variables
Example 2.14: Constant random variables
Example 2.15: Another example of a random variable: the identity mapping
Example 2.16: Creating new random variables from existing ones
Proposition 2.18: Measurability vs "level of information"
Example 2.19: Exploring the idea of distributions: the "behaviour" of random variables on the real line
Definition 2.20: The distribution of a random variable
Example 2.21: Different random variables with the same distribution
Proposition 2.22: Uniquely characterising distributions using cdf’s
Example 2.23: Working out a cdf
Example 2.25: A pdf is not unique
Lemma 2.26: Discrete vs continuous vs "other" random variables
Definition 2.27: The sigma-algebra generated by two (or more) functions/mappings
Proposition 2.28: Some properties of the sigma-algebra generated by two (or more) functions/mappings
Example 2.29: An example with three random variables
Definition 2.30: Independent sigma-algebras
Proposition 2.31: Independence of random variables vs the sigma-algebras they generate
Definition 2.33: Convergence of random variables: pointwise
Definition 2.34: Convergence of random variables: almost surely
Example 2.35: An example of almost sure convergence
Example 2.36: Another example of almost sure convergence
Definition 2.37: Convergence of random variables: in r-th mean/in L^r

Chapter 3:

Definition 3.1: What is an integral anyway? [⛔]
Proposition 3.2: Useful properties of positive and negative parts of a function
Example 3.3: Illustrating the role of positive and negative parts in integration
Example 3.4: Example of average value: constant functions
Example 3.5: Example of average value: piecewise constant functions
Example 3.6: Example of average value: linear functions
Proposition 3.7: The Riemann integral as the "average value" of a function
Definition 3.10: The Lebesgue integral of a non-negative step function
Proposition 3.11: Monotonicity of the Lebesgue integral of non-negative step functions
Definition 3.12: The Lebesgue integral of a non-negative function
Example 3.13: An example of a non-negative function with infinite Lebesgue integral
Definition 3.14: The Lebesgue integral of a non-negative function with infinity as possible value
Definition 3.16: The Lebesgue integral of any function
Example 3.17: The Lebesgue integral crash
Definition 3.18: Proper notation for the Lebesgue integral
Proposition 3.19: Lebesgue vs Riemann
Example 3.20: Lebesgue vs Riemann: Lebesgue wins!
Example 3.21: Examples of relationships between measures [⛔]
Definition 3.22: New definition of expectation of a random variable
Proposition 3.23: Consistency with old expressions for expectation
Theorem 3.24: Important properties of expectations
Theorem 3.25: Monotone Convergence Theorem (MCT)
Example 3.26: Simple example of the MCT resulting in infinities
Theorem 3.27: Dominated Convergence Theorem (DCT)
Theorem 3.29: Fatou’s Lemma [⛔]

Chapter 4:

Proposition 4.1: The "best prediction" for the value that a random variable takes is its expectation
Example 4.2: Motivation: conditioning in real life
Definition 4.3: Intuition behind conditioning on an event
Definition 4.4: Intuition behind conditioning on a random variable or sigma-algebra
Definition 4.5: Conditioning on an event: probabilities and expectations
Example 4.6: Simple example of conditioning on an event
Example 4.8: Simple example of conditioning on a sigma-algebra
Definition 4.9: Conditioning on a sigma-algebra
Definition 4.12: Conditioning on a random variable
Proposition 4.13: Conditioning on a random variable: backwards compatibility
Theorem 4.14: Properties of conditional expectations part 1: integral like ones
Theorem 4.15: Properties of conditional expectations part 2: concept specific ones

Chapter 5:

Example 5.1
Lemma 5.2
Example 5.3
Definition 5.4
Definition 5.6
Example 5.7
Definition 5.8
Proposition 5.9
Proposition 5.10
Theorem 5.12: Doob’s decomposition theorem
Example 5.13
Lemma 5.14
Lemma 5.15: Doob’s upcrossing lemma
Theorem 5.16: Doob’s martingale convergence theorem
Example 5.18

Chapter 6:

Definition 6.2
Example 6.3
Proposition 6.4
Theorem 6.5: Doob’s optional stopping theorem
Example 6.6
Theorem 6.7: Hunt’s optional stopping theorem
Lemma 6.9
Theorem 6.10: Doob’s martingale inequalities
Corollary 6.12: Doob’s martingale inequalities, the infinite horizon case

Chapter 7:

Lemma 7.1

Chapter 8:

Chapter 9:

Chapter 10: