I understand that the definition of a discrete-time martingale in a filtered probability space $(\Omega, F, \mathbb{P}, (F_n)_{n\geq 0} ) $ is $M = (M_n)_{n\geq 0}$ such that:
$M$ is adapted with respect to $(F_n)_{n\geq 0}$,
$M_n$ is intergrable for all $n \geq0$,
$M_N = E[M_{n+1}|F_n]$.
But how do you tell if a sequence of random variables $Y_N$ say, is adapted to $F_N$? I will provide a example to help understand my question.
Let $(\Omega, F, \mathbb{P}, ) $ be a probability space, where $(X_n)_{n\geq 1}$ is a sequence of independent random variables such that $X_N \sim LN(\mu, \sigma ^2).$ Let $F_0 := {(\emptyset, \Omega)}$ and let $F_N$ be the $\sigma$ - field generated by $X_1, ..., X_n$. $S_0 := 0$ and $S_n := \sum_{k = 1}^{n}{X_k}- an $.
Find $b \in \mathbb{R}$ such that $(Y_n)_{n\geq 0}$ defined as:
$Y_N := S_{n}^{2} - bn$.
Is a martingale with respect to the filtration $(F_n)_{n\geq 0}$.
How can I prove that $(Y_n)_{n\geq 0}$ is adapted to the filtration? I understand how to prove the other two conditions, if it has finite expectation it is integrable and then I can find out $b$ by using property (3).