def. An Outcome Space () is the set of all possible outcomes of an experiment def. An Event () is a subset of an outcome space. (die face is even)

Example. For a 6-sided die:

def. Probability for countable sets. When all outcomes in are equally likely, and is a finite set,

where denotes the number of elements in set (its cardinality).


  • Trivial cases:
  • is equivalent to (i.e. if A then B)

Measure Theory Definition

Probability is just an abuse of notation

Motivation. We need a sigma algebra to define a probability, because there’s problematic set theory problems with uncountably infinite sets.

def. Sigma Algebra. A sigma-algebra on a set denoted contains certain subsets of , such that it follows the following properties

  1. (biggest and smallest possible sigma-algebras)
  2. let be a subset of . If then . (closed under complement)
  3. let a subset of . If then (closed under finite union)

def. Measure. let . A measure is a function from measure space to to real numbers:

that satisfies the following properties:

  • Non-negative: ,
  • Countable Additivity:

Intuition. Consider a measure as a means to measure the “size” of the set (not Cardinality). On a real number line, the set has length . The set has length . Thus has length .

def. A Measure space is simply a set and its sigma-algebra together in a tuple:

Probability Space

def. A Probability measure on is a special type of measure that satisfies def. A Probability Space is like a measurable space, but also together with the probability function:

Interpretations of Probability (Philosophically)

Motivation. What is probability? Interpretations of its definitions seem meaningless without real-life experiments.

Two main interpretations of probability are:

  1. The Objectivist Interpretation—relative frequency of occurrence, if the experiment is conducted indefinitely
  2. The Subjectivist Interpretation—degree of belief; how much you would bet on an event.