stopping time

Hey, how long have I been sleeping? Sleepy smile, slightly embarrassed. Looking around for a clock or some hint from the sunlight.

"I'm not sure" -and I wasn't. I had not checked for a while and had been sitting on the other side of the room, half-awake, watching her sleep. I was thinking about where her dreams were taking her. Her breathing was slow and her eyelids moved in a soft motion
up,down

No, really, what time is it? , standing up now, trying to get her bearings. I forgot my watch so I kinda need to know.
"Really, I don't really know, I turned all the clocks off, the phone too."

Are the doors locked? Trying to look half serious, half kidding.
"Oh of course, I am just trying to make the day last longer. If you have to leave, that's all right."
She walked over slowly, sly smile and wraps arms around me and puts her head on my shoulder, yawns and blinks.
She is whispering in my ear:
Are you trying to make our time last longer?
"Of course" I say to her back, "Of course"
Well that's sweet, but I have to go.

I am not the first person to try this with her, I'm sure. The difference is that it worked for me. Time did stand still.

Oddly enough, to other people, your time will be the one that slows down. Go figure.

A stopping time is just that: think that you participate in a sequence of trials, the result of trial #i being X_i. You may stop the sequence at any time. Then obviously your decision to stop at time t depends only on the first t values.

Examples:

1. Stop a sequence of coin tosses at the second head.
2. Play a slot machine until either you lose all $100 you started off with, or you win $1000000.
3. Stop a sequence of die rolls as soon as more than half the rolls came up 6.
4. Employ a monkey to bang on a typewriter until it produces the complete text of Hamlet 2: The Prince is BACK.

Nonexamples:

1. Repeatedly bet double or quits on a fair coin toss stopping just before you lose.
2. Employ a monkey to bang on a typewriter until 30 minutes before it produces the complete text of Hamlet, then invite all your neighbours to see your educated monkey (bet on it producing Hamlet within 30 minutes).

As you can see, a stopping time is feasible, whereas other rules might not be. And, in fact, various theorems about betting are true for a fixed number of trials or for a stopping time (a fixed number is a stopping time!), but not for just any rule. Wald's theorem is probably the most important elementary result for stopping times.