In elementary probability you assign a number to every subset of a finite sample space. That breaks the moment the sample space becomes uncountable, like the unit interval or the real line. You cannot consistently assign a length to every subset of the reals; some pathological sets resist any sensible notion of size. The fix, due to Kolmogorov, is to restrict attention to a well-behaved collection of subsets called a sigma-algebra, and to measure only those. This collection is also the carrier of information, which is why filtrations matter so much in finance.
Think of a sigma-algebra as the catalog of questions you are allowed to ask. If you can ask whether an event happened, you can ask whether it did not happen, and if you can ask about countably many events you can ask whether at least one of them happened. Those closure rules are exactly the axioms below.
Formal definition
Let Ω be a nonempty set, the sample space. A collection F of subsets of Ω is a sigma-algebra if
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Lesson close · Measure-Theoretic Probability
Consolidate Sigma-Algebras & Probability Measures
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Key concepts covered
01Why we need more than a list of outcomes
02Formal definition
03The Borel sigma-algebra on the real line
04A short derivation: monotonicity from additivity
05Worked example
What you should know now
□Explain Sigma-Algebras & Probability Measures in your own words and state the assumptions.
□Reproduce the main argument, derivation, or implementation without copying the lesson.
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Check one example against units, boundary conditions, or edge cases.
□Name one modeling or implementation limitation and when it matters.
Sequence
Continue to Random Variables as Measurable Maps, the next prerequisite step in this course.
The pair (Ω,F) is a measurable space, and members of F are called measurable sets or events. From these axioms the empty set is measurable, since it equals Ωc, and countable intersections are measurable by De Morgan's law applied to complements of unions.
A probability measure is a function P:F→[0,1] satisfying Kolmogorov's axioms in measure-theoretic form:
P(Ω)=1,P(A)≥0,P(⋃n=1∞An)=∑n=1∞P(An) for pairwise disjoint An.
The last property is countable additivity, and it is strictly stronger than finite additivity; it is what makes limits behave. The triple (Ω,F,P) is a probability space.
The Borel sigma-algebra on the real line
We almost never want every subset of the reals, but we do want all the sets we can build from intervals. The Borel sigma-algebraB(R) is the smallest sigma-algebra containing every open interval. Smallest is well defined because an arbitrary intersection of sigma-algebras is again a sigma-algebra, so we may intersect all sigma-algebras that contain the open sets:
B(R)=σ({(a,b):a<b}).
This collection already contains closed sets, half-lines, single points, countable sets, and far more, yet it stays strictly smaller than the power set of the reals. Borel sets are the natural domain for probability on the line.
A short derivation: monotonicity from additivity
From the axioms alone we can derive that P is monotone. Suppose A⊆B, both in F. Write B as the disjoint union of A and B∖A, where B∖A=B∩Ac is measurable. Finite additivity, which follows from countable additivity by padding with empty sets, gives
P(B)=P(A)+P(B∖A)≥P(A),
since probabilities are nonnegative. The same disjoint-union trick yields the inclusion-exclusion identity and the continuity of measure along increasing or decreasing sequences of events.
Worked example
Let Ω={1,2,3,4} model a single roll of a four-sided die, and let F be the sigma-algebra generated by the single event even equals {2,4}. Closing under complement and union gives
F={∅,{2,4},{1,3},Ω}.
This four-element collection is the coarsest description in which you can tell even from odd but nothing finer. Assign P({2,4})=21. Then countable, here merely finite, additivity forces P({1,3})=21, P(∅)=0, and P(Ω)=1. Notice that the singleton {2} is simply not in F, so the question is this exactly two has no answer in this model. The sigma-algebra literally encodes the resolution of your information.
Filtrations as information
A filtration is an increasing family of sigma-algebras (Ft)t≥0 with Fs⊆Ft whenever s≤t. Each Ft is the catalog of events whose occurrence is known by time t. Because the family only grows, information is never lost. A price process observed through time generates exactly such a filtration, and the requirement that a trading position depend only on currently known events is the formal meaning of no looking into the future.
Why quants care
Every arbitrage-pricing argument rests silently on a probability space and a filtration. Adapted processes, martingales, and conditional expectations are all defined relative to the chosen Ft, so the sigma-algebra you pick is the precise statement of what a trader can observe and act upon. Change of measure, the engine behind risk-neutral pricing, replaces P by an equivalent measure Q on the sameF, which is only coherent because measures live on sigma-algebras. Getting these foundations right is what separates a rigorous derivation of a hedging strategy from a hand-wave.
Adapted from MIT OpenCourseWare (18.175 Theory of Probability), CC BY-NC-SA.
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Problems
P1
Let Ω={1,2,3,4,5,6} and let F be the sigma-algebra generated by the single set A={1,2}. How many distinct sets does F contain?
P2
Using only the Kolmogorov axioms, prove that for any events A,B∈F we have P(A∪B)=P(A)+P(B)−P(A∩B).
Knowledge check
Q1.Which property distinguishes a probability measure from a merely finitely additive set function?
Q2.The Borel sigma-algebra on the real line is best described as:
Q3.In a filtration, the inclusion Fs⊆Ft for s≤t formalizes which idea?
Q4.Why can we not simply take the power set of the reals as our event collection for probability?
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