Independence of sets of sets and measure spaces (σ-algebras) #

• A family of sets of sets π : ι → set (set α) is independent with respect to a measure μ if for any finite set of indices s = {i_1, ..., i_n}, for any sets f i_1 ∈ π i_1, ..., f i_n ∈ π i_n, μ (⋂ i in s, f i) = ∏ i in s, μ (f i). It will be used for families of π-systems.
• A family of measurable space structures (i.e. of σ-algebras) is independent with respect to a measure μ (typically defined on a finer σ-algebra) if the family of sets of measurable sets they define is independent. I.e., m : ι → measurable_space α is independent with respect to a measure μ if for any finite set of indices s = {i_1, ..., i_n}, for any sets f i_1 ∈ m i_1, ..., f i_n ∈ m i_n, then μ (⋂ i in s, f i) = ∏ i in s, μ (f i).
• Independence of sets (or events in probabilistic parlance) is defined as independence of the measurable space structures they generate: a set s generates the measurable space structure with measurable sets ∅, s, sᶜ, univ.
• Independence of functions (or random variables) is also defined as independence of the measurable space structures they generate: a function f for which we have a measurable space m on the codomain generates measurable_space.comap f m.

Main statements #

• TODO: Indep_of_Indep_sets: if π-systems are independent as sets of sets, then the measurable space structures they generate are independent.
• indep_of_indep_sets: variant with two π-systems.

Implementation notes #

We provide one main definition of independence:

• Indep_sets: independence of a family of sets of sets pi : ι → set (set α). Three other independence notions are defined using Indep_sets:
• Indep: independence of a family of measurable space structures m : ι → measurable_space α,
• Indep_set: independence of a family of sets s : ι → set α,
• Indep_fun: independence of a family of functions. For measurable spaces m : Π (i : ι), measurable_space (β i), we consider functions f : Π (i : ι), α → β i.

Additionally, we provide four corresponding statements for two measurable space structures (resp. sets of sets, sets, functions) instead of a family. These properties are denoted by the same names as for a family, but without a capital letter, for example indep_fun is the version of Indep_fun for two functions.

The definition of independence for Indep_sets uses finite sets (finset). An alternative and equivalent way of defining independence would have been to use countable sets. TODO: prove that equivalence.

Most of the definitions and lemma in this file list all variables instead of using the variables keyword at the beginning of a section, for example lemma indep.symm {α} {m₁ m₂ : measurable_space α} [measurable_space α] {μ : measure α} ... . This is intentional, to be able to control the order of the measurable_space variables. Indeed when defining μ in the example above, the measurable space used is the last one defined, here [measurable_space α], and not m₁ or m₂.

References #

• Williams, David. Probability with martingales. Cambridge university press, 1991. Part A, Chapter 4.

Deducing indep from Indep #

π-system lemma #

Independence of measurable spaces is equivalent to independence of generating π-systems.

Independence of measurable space structures implies independence of generating π-systems #

Independence of generating π-systems implies independence of measurable space structures #

Independence of measurable sets #

We prove the following equivalences on indep_set, for measurable sets s, t.

• indep_set s t μ ↔ μ (s ∩ t) = μ s * μ t,
• indep_set s t μ ↔ indep_sets {s} {t} μ.