M.3.2 Probability Spaces & Filtrations

Basic Prerequisites: 01 · From Zero.

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1. Intuition & Practical Objective

A $\sigma$-algebra is information written as a set of questions you can answer Yes/No. At time $t$ you know the past of the stock price but not its future; the collection of events you can distinguish is ${\mathcal{F}}_t$, and a filtration ${\mathcal{F}}_0\subseteq {\mathcal{F}}_1\subseteq \cdots \subseteq \mathcal{F}$ is those sets growing as time passes and information accumulates. A process is adapted if it only ever uses today’s-and-earlier information - which is the mathematical statement of “no lookahead bias.”

The practical objective of this page: understand the three linked ideas - (1) $\sigma$-algebras as the closed family of measurable events; (2) the probability measure with its countable additivity; (3) filtrations and ${\mathcal{F}}_t$-measurability as the exact language of “information available at time $t$.” Everything downstream - conditional expectation, martingales, no-arbitrage - is defined relative to a filtration, so getting this precise is the hinge of the whole Atlas.

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2. Mathematical Ground Truth & Derivations

2.1 $\sigma$-algebra (Shreve II Def 1.1.1; Shreve I §2.2)

On nonempty $\Omega$, a family $\mathcal{F}$ of subsets is a $\sigma$-algebra if: $\varnothing \in \mathcal{F}$; $A\in \mathcal{F}\Rightarrow A^c\in \mathcal{F}$; $A_1,A_2,\cdots \in \mathcal{F}\Rightarrow {\bigcup }_n{A}_n\in \mathcal{F}$. Closure under countable unions + complements gives countable intersections, finite unions, and $\Omega \in \mathcal{F}$. The Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ is the smallest $\sigma$-algebra containing the open sets; a random variable is $X:\Omega \to \mathbb{R}$ with $X^{-1}(B)\in \mathcal{F}$ for every Borel $B$ (Shreve I Ch 11). On a finite space (e.g. $n$ coin tosses) one takes $\mathcal{F}=$ all $2^{2^n}$-many subsets, and the theory reduces to counting.

2.2 Probability measure & countable additivity (Shreve II Def 1.1.2)

$\mathbb{P}:\mathcal{F}\to [0,1]$ with $\mathbb{P}(\Omega )=1$ and, for disjoint $A_n$,

$$\mathbb{P}\left(\bigcup _n{A}_n\right)=\sum _n\mathbb{P}(A_n).$$

Finite additivity and $P(A^c)=1-P(A)$ follow. On $[0,1]$, uniform/Lebesgue measure starts from $\mathbb{P}(a,b]=b-a$ on intervals and extends to Borel sets by countable additivity (Shreve II Ex 1.1.3).

2.3 The generated $\sigma$-algebra: what does a random variable “reveal”? (Shreve I §2.2; Shreve II §2.1)

The natural filtration on the coin-toss space is ${\mathcal{F}}_k=\sigma (S_1,\dots ,S_k)$, the smallest $\sigma$-algebra w.r.t. which $S_1,\dots ,S_k$ are measurable. Its atoms are the sets of paths sharing the first $k$ tosses: ${\mathcal{F}}_1$ has two atoms (first toss H or T), ${\mathcal{F}}_2$ has four, etc. $X$ is ${\mathcal{F}}_k$-measurable iff its value is determined by the first $k$ tosses (Shreve I §2.2). This is exactly “information revealed by the stock up to time $k$.“

2.4 Filtrations, adaptedness, and the flow of time (Shreve I §2.4; Shreve II Def 3.3.3)

A filtration is an increasing family ${\mathcal{F}}_0\subseteq {\mathcal{F}}_1\subseteq \cdots \subseteq {\mathcal{F}}_n$; a process $\{M_k\}$ is adapted if $M_k$ is ${\mathcal{F}}_k$-measurable for each $k$. For continuous Brownian motion the filtration satisfies (Shreve II Def 3.3.3) $W$ adapted and future increments $W(u)-W(t)$ independent of $\mathcal{F}(t)$ for $u\ge t$ - the efficient-markets formulation: today’s information tells you nothing about the direction of tomorrow’s move.

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3. Computational Implementation - the coin-toss space with its filtration

Build the 3-toss space, compute $\mathbb{E}[X\mid {\mathcal{F}}_1]$ by averaging over atoms (the finite-space reduction of partial averaging), and verify partial averaging, the tower property, and ${\mathcal{F}}_1$-measurability. Stdlib only.

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4. Failure Modes & First-Principles Breakdowns

1. Lookahead bias = ${\mathcal{F}}_t$-violation. A backtest feature computed with bar $t$‘s close $C_t$ before that close has occurred is ${\mathcal{F}}_{t+\Delta t}$-measurable, not ${\mathcal{F}}_t$-measurable - it peeks one bar into the future. Root cause: the feature is not adapted to the trading filtration. (Shreve I §2.2.)
2. Believing every subset is measurable. On an uncountable $\Omega$ not every set is in $\mathcal{F}$; $\mathcal{B}(\mathbb{R})$ and Lebesgue measure are built on Borel sets for exactly this reason (Shreve II Ex 1.1.3). A “probability” that assigns mass to a non-measurable set is simply undefined.
3. Requiring a measure on outcomes. Since ${\Omega }_{\infty }$ is uncountable, single-path probabilities are forced to $0$; demanding $\mathbb{P}(\{\omega \})>0$ for each $\omega$ makes $\mathbb{P}(\Omega )=\infty$, breaking normalization. The measure must live on sets (Shreve II §1.1).
4. Forgetting the filtration when defining a martingale/expectation. "$\mathbb{E}[M_{t+1}]=M_t$" is false in general (expectations are numbers); the correct statement is conditional on ${\mathcal{F}}_t$. Dropping the filtration turns the definition into nonsense.

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5. References

• Shreve, Stochastic Calculus for Finance II
• Shreve, Stochastic Calculus for Finance I
• Shreve, Stochastic Calculus for Finance I
• Williams, Probability with Martingales

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6. Connected Graph Bridges

• Back: 01 · From Zero · Index Hub
• Forward: 03 · Distributions & Expectation · 04 · Conditional Expectation
• Theory: Stochastic Calculus (BM filtration Def 3.3.3) · No-Arbitrage & Binomial Trees (discrete filtration ${\mathcal{F}}_k$)