In my lecture slides, there is this example (which is left as an exercise) that I am not quite sure about the answer:
Suppose that $(B_n)_{n\in\mathbb{N^+}}$ be a sequence of rv's with finite expectation and define recursively, $Y_1=B_1$ and: $$Y_n = B_n - \mathbb{E}[B_n|B_1,..,B_{n-1}]+Y_{n-1}$$ for $n\geq2.$ Show that $(Y_n)_{n\in\mathbb{N^+}}$ is a martingale, and specify the relevant history.
My approach to this question is to first define the history $\mathcal{F_n}$ as $B_1,...,B_n$.
Then we take expectation both sides: $$\mathbb{E}[Y_n|\mathcal{F}_{n-1}]= \mathbb{E}[B_n|\mathcal{F}_{n-1}]-\mathbb{E}[\mathbb{E}[B_n|\mathcal{F}_{n-1}]|\mathcal{F}_{n-1}]+\mathbb{E}[Y_{n-1}|\mathcal{F}_{n-1}]$$
By the law of iterated expectation $\mathbb{E}[\mathbb{E}[B_n|\mathcal{F}_{n-1}]|\mathcal{F}_{n-1}]= \mathbb{E}[B_n|\mathcal{F}_{n-1}]$, which gives: $$\mathbb{E}[Y_n|\mathcal{F}_{n-1}]= \underbrace{\mathbb{E}[B_n|\mathcal{F}_{n-1}]-\mathbb{E}[\mathbb{E}[B_n|\mathcal{F}_{n-1}]|\mathcal{F}_{n-1}]}_\text{0}+\mathbb{E}[Y_{n-1}|\mathcal{F}_{n-1}]$$
Hence, $\mathbb{E}[Y_n-Y_{n-1}|\mathcal{F}_{n-1}]= 0$, which looks like a Martingale Difference Sequence. But does this imply $(Y_n)_{n\in\mathbb{N^+}}$ is a martingale relative to history $\mathcal{F}_n$? I understand that if $M_t$ is a martingale then $\mathbb{E}(M_t-M_{t-1}|\mathcal{F}_{t-1})=0$, but have not seen the proof of the converse.
If it is not, how do I solve this problem?
For the converse, you need to show that $M_t$ is $\mathcal F_t$-measurable. In your context, you can show by induction that the random variable $Y_n$ is $\mathcal F_n$-measurable.