stopping time definitions Let $\tau$ a stopping time w.r.t. the filtration $(F_n)_{n\in{\mathbb{N}}}$. The stopped $\sigma$-field $F_\tau$ associated with $\tau$ is defined to be
$F_\tau$:=$\{A\in{F}$$:A\cap{\{\omega:\tau(\omega)=n\}\in{F_n} \text{ for all }n\geq {0}}\}$
I know this definition is equivalent to
$F_\tau$:=$\{A\in{F}$$:A\cap{\{\omega:\tau(\omega)\leq{n}\}\in{F_n} \text{ for all }n\geq {0}}\}$
But is it the same as
$F_\tau$:=$\{A\in{F}$$:A\cap{\{\omega:\tau(\omega)<{n}\}\in{F_n} \text{ for all }n\geq {0}}\}$?
I have a feeling the answer is no, but I don't know how to start disproving it. Any help is appreciated! Thanks!:D
 A: Indeed, the answer is no.  
Let $X_1, X_2$ be independent flips of a coin, with $P(X_k=H)=P(X_k=T)=1/2$ for both.  Let $\mathcal F_1 = \sigma(X_1)$,
$\mathcal F_2 = \mathcal F = \sigma(X_1,X_2)$.  Define $\tau = 1$ if $X_1=H$, and $\tau = 2$ otherwise.  Then $\tau$ is a stopping time: we know at time $1$ whether $\tau = 1$ and we know at time $2$ whether $\tau=2$.  
Take $A=\{X_1=H, X_2=H\}$, an event of probability $1/4$.  Of course $A \not\in \cal F_\tau$ since
$$
A \cap \{\tau=1\} = A \not\in \cal F_1 .
$$
We don't know until time 2 whether $A$ occurs, so if $\tau=1$ and we reach time $1$, we don't know whether $A$ occurs:  $A$ is not an event known at time $\tau$.
But $A$ does satisfy your last criterion:
$$\begin{align}
A \cap \{\tau < 1\} &= \varnothing \in \cal F_1\\
A \cap \{\tau < 2\} &= A \in \cal F_2
\end{align}$$
remark 
Perhaps
$$
A\cap{\{\omega:\tau(\omega)=n\}\in{\mathcal F_n} \text{ for all }n\geq {0}}
$$
is the most natural definition for $\mathcal F_\tau$.  But the equivalent one
$$
A\cap{\{\omega:\tau(\omega)\le n\}\in{\mathcal F_n} \text{ for all }n\geq {0}}
$$
must be used when we study continuous-time stochastic processes.  In that situation we have stochastic basis $\cal F_t$ indexed by a real parameter $t$.  And it is often the case that all events $\{\tau = t\}$ have probability zero.
