The mathematics of measure formalizes the intuitive notion of probability, associating an event with a set of outcomes and defining the probability of the event to be its volume or measure relative to that of a universe of possible outcomes[ref]Glassermann, Paul. Monte Carlo Methods in Financial Engineering. Chapter 1.1.1.[/ref]

You may wish to review sets and mappings in any suitable textbook although this is not absolutely necessary. The material covered here is of necessity abstract. It defines the general framework within which we measure the real (or imaginary), mathematical (or statistical) constructs that are fundamental to pricing theory and mathematical finance.

Measure Space

A measure space is a triple ($X$, $\mathcal{F}$, $\mu$), where $\mu$ is a measure on the measurable space ($X$,$\mathcal{F}$)[ref]Björk, Tomas. Arbitrage Theory in Continuous Time, third edition. Appendix A, Definition A10.[/ref]

Imagine two (or more) fist sized lumps of clay on a pottery wheel, about to be spun into a bowl or vase. The potter could have a lump of clay in the center of the wheel and another lump near the edge. Each of these lumps is a subset $A_{1,2\dots n}$ within the measurable space of the wheel $\Omega$. We say $A \subseteq \Omega$ and call $\Omega$ the sample space and it represents the entirety of ‘possible states of the world’. I this case the pottery wheel is the entire world of the potter.

These two lumps of clay are “nice” measurable clay (can be weighed on a scale), unlike unusable clay residue on the wheel. Altogether, the measurable clay $A$ is called the sigma algebra $\mathcal{F}$ and it represents ‘sets of states of the world’. (Think: fistfuls of clay on the wheel).

$P$ is the probability of an event. The probability that all the clay on the wheel will end up as a vase? As this depends somewhat on how much clay there is it is logical to say that the event $E \in \mathcal{F}$ meaning the event is an element of the sigma algebra. All clay on the wheel will end up as a clay item (pot, vase bowl etc). We therefore can say

$P({\Omega}) = 1$We assume that because of gravity small lumps of clay on the periphery of the wheel fall to the ground. (The periphery is unusable surface area of the sample space $\Omega$). So we identify clay on the periphery of the wheel as the impossible event (the impossible amount of clay in the total amount of clay $A$ ). Technically this the impossible event $\emptyset$ in $\mathcal{F}$.

The measure space we are interested in is a specific measure space called a probability space ($\Omega$, $\mathcal{F}$, $P$) comprised of sample space $\Omega$, sigma algebra $\mathcal{F}$ and probability $P$ which lies in $\mathcal{F}$.

Filtered Probability Space

Take ($\Omega$, $\mathcal{F}$, $P$, $\underline{\mathcal{F}}$) as a filtered probability space. Let a random process $X$ be defined on this space in continuous or discrete time.

Process $X$ is an $\underline{\mathcal{F}}$ martingale if