Questions about the concept of strong Markov property I am trying to understand the concept of strong Markov property quoted from Wikipedia:

Suppose that $X=(X_t:t\geq 0)$ is a
  stochastic process on a probability
  space
  $(\Omega,\mathcal{F},\mathbb{P})$ with
  natural filtration
  $\{\mathcal{F}\}_{t\geq 0}$. Then $X$
  is said to have the strong Markov
  property if, for each stopping time
  $\tau$, conditioned on the event
  $\{\tau < \infty\}$, the process
  $X_{\tau + \cdot}$ (which maybe needs
  to be defined) is independent from
  $\mathcal{F}_{\tau}:=\{A \in  \mathcal{F}: \tau \cap A \in  \mathcal{F}_t ,\, \ t \geq 0\}$ and
  $X_{\tau + t} − X_{\tau}$ has the same
  distribution as $X_t$ for each $t \geq  0$.

Here are some questions that make me stuck:


*

*In $\mathcal{F}_{\tau}:=\{A \in
    \mathcal{F}: \tau \cap A \in
    \mathcal{F}_t ,\, \ t \geq 0\} $,
what does $\tau \cap A $ mean?
$\tau$ is a stopping time and
therefore a random variable and $A$
is a $\mathcal{F}$-measurable
subset, but what does $\tau \cap A$
mean?

*How is the process $X_{\tau + \cdot}$ defined from the process $X_{\cdot}$ ? Is it the translated
version of the latter by $\tau$?

*How is the conditional independence
between a process, such as $X_{\tau
    + \cdot}$,  and the sigma algebra, such as $\mathcal{F}_{\tau}$, given
an event, such as $\{\tau <
    \infty\}$, defined? 
Related question, is independence
between a random variable and a
sigma algebra defined as
independence between the sigma
algebra of the random variable and
the sigma algebra?

*Is "$X_{\tau+ t}  − X_{\tau}$ has the
same distribution as $X_t$ for each
$t \geq 0$" also conditional on the
event $\{\tau < \infty\}$?


Thanks and regards!
 A: For the first one the notation is wrong it should I think $A\cap\{\tau\le t\}\in\mathcal{F}_t$ instead of $A\cap\tau$.  
And for the fourth point, look at George Lowther comments below, that fully address the problematic.
Regard
A: Here is a less garbled version of the Wikipedia definition. (Use TheBridge's correction for the definition of ${\cal F}_\tau$.) 
  The post-$\tau$ process $X_{\tau+\cdot}$ is defined on the event $\{\tau<\infty\}$ by
$$
X_{\tau+t}(\omega) = X_{\tau(\omega)+t}(\omega),\qquad t\ge 0,
$$
for  $\omega\in\{\tau<\infty\}$. One way to state the strong Markov property is this: The conditional distribution of $X_{\tau+\cdot}$ given ${\cal F}_\tau$ is (a.s.) equal to the conditional distribution of 
$X_{\tau+\cdot}$ given $\sigma\{X_\tau\}$, on the event $\{\tau<\infty\}$. More precisely,
$$
P[ X_{\tau+t}\in B|{\cal F}_\tau] = P[ X_{\tau+t}\in B|X_\tau],\qquad \hbox{almost surely on }\{\tau<\infty\},
$$
for all $t\ge 0$, and all measurable subsets $B$ of the state space of $X$.
This is equivalent to the statement that $X_{\tau+\cdot}$ and ${\cal F}_\tau$ are conditionally independent, given $X_\tau$:
$$
P[ F\cap \{X_{\tau+t}\in B\}|X_\tau] = P[ F|X_\tau]\cdot P[X_{\tau+t}\in B|X_\tau],\qquad \hbox{almost surely on }\{\tau<\infty\},
$$
