In the book "Optimal Stopping and Free-Boundary Problems" there is given Doob's inequality of the following form.
Let $X = (X_t,F_t)$ be a submartingale. Then for any $\varepsilon>0$ and each $T>0$ $ \mathsf P\left\{\sup\limits_{t\leq T}|X_t|\geq \varepsilon\right\}\leq \frac 1\varepsilon \sup\limits_{t\leq T}\,\,\,\mathsf E\,|X_t|. $
On the other hand, George Lowther presented the following example. Let $X_0 = 0$, $ X_1 = \begin{cases} 1, &\quad p = 1/3; \\ 0, &\quad p = 1/3; \\ -1,&\quad p =1/3. \end{cases} $ and $X_2 = 1$ if $X_1 = 1$, $X_2 = 0$ if $X_1 = -1$ and
X_2 = \begin{cases} 1, &\quad p = 1/2;
\ -1,&\quad p =1/2. \end{cases} if $X_1 = 0$. Finally, $X_n = X_2$ for all $n\geq 2$.
It is easy to check that this is a submartingale. On the other hand, for $\varepsilon = 1$ we have $ \mathsf P\left\{\sup\limits_{n\leq 2}|X_n|\geq 1\right\} = 1 $ but $\mathsf E[|X_0|] = 0$, $\mathsf E[|X_1|] = 2/3$ and $\mathsf E[|X_2|] = 2/3$ - so we have inequality $1\leq 2/3$.
Could you help with finding a mistake since I am confused?
The reference can be seen here: http://books.google.com/books?id=UinZbLqpUDEC&pg=PA60&source=gbs_toc_r&cad=4#v=onepage&q&f=false page 62.
This question raised from the discussion here: Bounds for submartingale