TITLE: Interpretation of sigma algebra
QUESTION [30 upvotes]: My question is how to interpret sigma algebra, especially in the context of probability theory (stochastic processes included). I would like to know if there is some clear and general way to interpret sigma algebra, which can unify various ways of saying it as history, future, collection of information, size/likelihood-measurable etc? 
Specifically,I hope to know how to interpret the following in some consistent way:

being given/conditional on a sigma algebra
a subset being measurable or nonmeasurable w.r.t. a sigma
algebra
a mapping being measurable or nonmeasurable w.r.t. a
sigma algebra in domain and another
sigma algebra in codomain
a collection of increasing sigma algebras, i.e. a filtration of sigma algebras
...

Following are a list of examples that I have met. They are nice examples, but I feel their ways of interpretation are not clear and consistent enough for me to apply in practice. Even if there is no unified way to interpret all the examples, I would like to know what some different ways of interpretation are. 

Stopping time

Let $(I, \leq)$ be an ordered index
  set, and let $(\Omega, \mathcal{F},\mathcal{F}_t, \mathbb{P})$ be a
  filtered probability space. 
Then a random variable $\tau : \Omega \to I$ is called a stopping time if
  $\{ \tau \leq t \} \in \mathcal{F}_{t} \forall t \in I$. 
Speaking concretely, for τ to be a
  stopping time, it should be possible
  to decide whether or not $\{ \tau \leq t \}$ has occurred on the basis of the
  knowledge of $\mathcal{F}_t$, i.e.,
  event $\{ \tau \leq t \}$ is
  $\mathcal{F}_t$-measurable.

I was still wondering how exactly to "decide whether or not $\{ \tau \leq t \}$ has occurred on the basis of the knowledge of $\mathcal{F}_t$, i.e., event $\{ \tau \leq t \}$ is  $\mathcal{F}_t$-measurable." 
Martingale process 

If a stochastic process $Y : T \times \Omega \rightarrow S$ is a martingale
  with respect to a filtration $\{ \Sigma_t\}$ and probability measure
  $P$, then  for all s and t with $s < t$ and all $F \in \Sigma_s$,
          $$Y_s = \mathbf{E}_{\mathbf{P}} ( Y_t | \Sigma_s ),$$ 

where $\Sigma_s $ is interpreted as "history". 
I was also wondering how $\Sigma_s, s < t$ can act as history, $\Sigma_s, s=t$ as present, and $\Sigma_s, s > t$ as future?
I originally interpret a measurable
subset wrt a sigma algebra as a
subset whose "size"/"likelihood"  is measurable,
and the class of such
size-measurable subsets must be
closed under complement and
countable union.
In a post by Nate Eldredge, a
measurable subset wrt a sigma
algebra is interpreted by analogy of questions being answered:

If I know the answer to a question
  $A$, then I also know the answer to
  its negation, which corresponds to the
  set $A^c$ (e.g. "Is the dodo
  not-extinct?").  So any information
  that is enough to answer question $A$
  is also enough to answer question
  $A^c$.  Thus $\mathcal{F}$ should be
  closed under taking complements. 
  Likewise, if I know the answer to
  questions $A,B$, I also know the
  answer to their disjunction $A \cup B$
  ("Are either the dodo or the elephant
  extinct?"), so $\mathcal{F}$ must also
  be closed under (finite) unions. 
  Countable unions require more of a
  stretch, but imagine asking an
  infinite sequence of questions
  "converging" on a final question. 
  ("Can elephants live to be 90? Can
  they live to be 99? Can they live to
  be 99.9?" In the end, I know whether
  elephants can live to be 100.)


Thanks in advance for sharing your views, and any reference that has related discussion is also appreciated!

REPLY [2 votes]: $\mathcal{F}_0=\{\emptyset,\Omega\}$
$\mathcal{F}_1=\sigma(\{HH,HT\},\{TH,TT\})=\{\emptyset, \Omega, \{HH,HT\},\{TH,TT\}\}\supset \mathcal{F}_0$
$\mathcal{F}_2=\sigma(\{HH\},\{HT\},\{TH\},\{TT\})=\{\emptyset, \Omega, \{HH\},\{HT\},\{TH\},\{TT\},\{HH,HT\},\{HH,TH\},\{HH,TT\},\{HT,TH\},\{TH,TT\},\{HT,TT\},\{HH,TH,HT\},\{HH,HT,TT\},\{HH,TH,TT\},\{HT,TH,TT\}\}\supset \mathcal{F}_1\supset \mathcal{F}_0$