A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set ${\displaystyle \Omega }$ satisfying a set of axioms weaker than those of 𝜎-algebra. Dynkin systems are sometimes referred to as 𝜆-systems (Dynkin himself used this term) or d-system. These set families have applications in measure theory and probability.

A major application of 𝜆-systems is the π-𝜆 theorem, see below.

Let ${\displaystyle \Omega }$ be a nonempty set, and let ${\displaystyle D}$ be a collection of subsets of ${\displaystyle \Omega }$ (that is, ${\displaystyle D}$ is a subset of the power set of ${\displaystyle \Omega }$). Then ${\displaystyle D}$ is a Dynkin system if

It is easy to check that any Dynkin system ${\displaystyle D}$ satisfies:

Conversely, it is easy to check that a family of sets that satisfy conditions 4-6 is a Dynkin class. For this reason, a small group of authors have adopted conditions 4-6 to define a Dynkin system.

An important fact is that any Dynkin system that is also a π-system (that is, closed under finite intersections) is a 𝜎-algebra. This can be verified by noting that conditions 2 and 3 together with closure under finite intersections imply closure under finite unions, which in turn implies closure under countable unions.

Given any collection ${\displaystyle {\mathcal {J}}}$ of subsets of ${\displaystyle \Omega ,}$ there exists a unique Dynkin system denoted ${\displaystyle D\{{\mathcal {J}}\}}$ which is minimal with respect to containing ${\displaystyle {\mathcal {J}}.}$ That is, if ${\displaystyle {\tilde {D}}}$ is any Dynkin system containing ${\displaystyle {\mathcal {J}},}$ then ${\displaystyle D\{{\mathcal {J}}\}\subseteq {\tilde {D}}.}$ ${\displaystyle D\{{\mathcal {J}}\}}$ is called the Dynkin system generated by ${\displaystyle {\mathcal {J}}.}$ For instance, ${\displaystyle D\{\varnothing \}=\{\varnothing ,\Omega \}.}$ For another example, let ${\displaystyle \Omega =\{1,2,3,4\}}$ and ${\displaystyle {\mathcal {J}}=\{1\}}$; then ${\displaystyle D\{{\mathcal {J}}\}=\{\varnothing ,\{1\},\{2,3,4\},\Omega \}.}$

Sierpiński-Dynkin's π-𝜆 theorem: If ${\displaystyle P}$ is a π-system and ${\displaystyle D}$ is a Dynkin system with ${\displaystyle P\subseteq D,}$ then ${\displaystyle \sigma \{P\}\subseteq D.}$