I am studying the Doob Meyer Decomposition theorem. The resource I am following is Prof.Phillip Protter's textbook:Stochastic Integration and Differential Equations.
My questions relate to one of the lemmas in the Doob-Meyer Decompositions section on page 108. The statement of this lemma is:(or you could find this lemma on Prof. Richard Bass's paper: 'The Doob-Meyer Decomposition Revisited'.lemma 2.5 on page 4)
Choose and fix a $\nu\in\mathbb{Z}_+$, and let $D_n = \{k2^{-n}: 0\leq k2^{-n}\leq \nu\}$
$\bf{Lemma 2.}$ Let $T$ be a totally inaccessible stopping time. For $\delta > 0$, let $R(\delta) = \sup_{t\leq\nu} P(t\leq T\leq t+\delta|\mathcal{F}_t)$. Then $R(\delta)\rightarrow 0$ in probability as $\delta\rightarrow 0$.
I list the following three questions:
(1) Within the proof, it defines the following stopping times $$S_n(\delta) = \inf_{t}\{t\in D_n, P(t\leq T\leq t+\delta|\mathcal{F}_t)>a\}\wedge\nu$$
$$\bar{S}(\delta) = \inf_{n} S_n(\delta)$$ $$S = \sup_n \bar{S}(\frac{1}{n})$$ I couldn't see why $S$ here is accessible on $\{S =T\}$.
(2)It is not very clear to me why the inequality on the top of page 109 holds: for large n, $$P(E\{\mathbb{1}_{S_n(\delta)\leq T\leq S_n(\delta)+\delta}|\mathcal{F}_{S_n(\delta)}\} > a)\geq \epsilon$$ My thought on this is, it requires some sort of a.s right continuity of $f(t) :=E\{\mathbb{1}_{t \leq T\leq t+\delta}|\mathcal{F}_{t}\}$, but I don't see why this is true.
(3)Based on Richard Bass' paper as I mentioned above(proof of lemma 2.5 on page 4), it claims $$\text{from }P(E\{\mathbb{1}_{S_n(\delta)\leq T\leq S_n(\delta)+\delta}|\mathcal{F}_{S_n(\delta)}\} > a)\geq \epsilon\text{ to }P(E\{\mathbb{1}_{\bar{S}(\delta)\leq T\leq \bar{S}(\delta)+\delta}|\mathcal{F}_{\bar{S}(\delta)}\} > a)\geq \epsilon$$
and
$$\text{from }P(E\{\mathbb{1}_{\bar{S}(\delta)\leq T\leq \bar{S}(\delta)+\beta}|\mathcal{F}_{\bar{S}(\delta)}\} > a)\geq \epsilon\text{ to }P(E\{\mathbb{1}_{S\leq T\leq S+\beta}|\mathcal{F}_{S-}\} > a)\geq \epsilon\text{ to }$$ it use the remark 2.4 twice. This remark also appears as the lemma on page 107 in the Phillip Protter's book. It is also not very clear to me how this remark is helpful making the above two arguments.
Since I don't understand the proof very well, I have to apologize that my questions maybe not specific enough and unorganized. Please feel free to leave comments and thank you very much!