In Markov Chains, how to prove that a random variable belongs to the “future”?

I am trying to prove the Exercise 2.15 from the book “Markov Chains”, by Daniel Revuz:

A real random variable ZZ is σ(Xm,m≥n)\sigma (X_m, m \geq n)-measurable if and only if Z=Z′∘θnZ = Z’ \circ \theta^n, where Z′∈FZ’ \in \mathscr{F}.

The definition of σ(Xm,m≥n)\sigma(X_m,m\geq n) is the smallest σ\sigma-algebra such that every random variable Xm,m≥nX_m, m\geq n is measurable. But everywhere I look for, the authors always refer to σ(Xm,m≥n)\sigma(X_m,m\geq n) as the “future” (Definition 2.1).


θn\theta^n is the nn-left-shift operator on a sequence:

F\mathscr{F} is a σ\sigma-algebra on the space of sequences.

Proof (my try):

If ZZ is measurable with respect to σ(Xm,m≥n)\sigma(X_m,m\geq n) then ZZ does not depend on the first nn coordinates, so:

Define Z′(gk):=Z(e,e,…,g0,g1,…)Z'(g_k):=Z(e,e,\ldots,g_0,g_1,\ldots) where g0g_0 is on the nn-th coordinate, then:

Conversely, suppose that there exists a random variable Z′Z’ such that Z=Z′∘θnZ=Z’\circ\theta^n. Let us show that ZZ does not depend on the first nn coordinates (Is this a sufficient condition? I couldn’t prove that ZZ is measurable with respect to σ(Xm,m≥n)\sigma(X_m,m\geq n) using inverse images).

Let (gk),(hk)∈Ω(g_k),(h_k)\in\Omega such that gk=hkg_k=h_k for all k≥nk\geq n, then