Sets of measure 0 that are not empty or are even infinite are indeed mind-bending when first encountered. But a set of cardinality 1 may be fun, too.
Take a real-valued stochastic processbS, and an arbitrary time t. The process generates a value, v = S(t).
Since the width of the range from v to v is zero, the probability that S generates this number is also zero.
We can replace it with a physical stochastic process. Say, take the voltage in your electric socket. It fluctuates randomly around the nominal value. At every moment there is some voltage in the socket, and the probability to encounter this voltage is zero. It looks like the impossible is actually happening at every moment!
This is, of course, thevprobability being a chance to encounter a given value. If we pick a value first, and then start sampling our stochastic process, there's zero probability that we're going to encounter our (infinitely precise) number, no matter how long we'd try.
(A physical voltmeter has a finite resolution, say, 0.10 V, so it's going to show you a predetermined voltage pretty often. But it would be a rounded approximation of the unpredictable real value.)
Take a real-valued stochastic processbS, and an arbitrary time t. The process generates a value, v = S(t).
Since the width of the range from v to v is zero, the probability that S generates this number is also zero.
We can replace it with a physical stochastic process. Say, take the voltage in your electric socket. It fluctuates randomly around the nominal value. At every moment there is some voltage in the socket, and the probability to encounter this voltage is zero. It looks like the impossible is actually happening at every moment!
This is, of course, thevprobability being a chance to encounter a given value. If we pick a value first, and then start sampling our stochastic process, there's zero probability that we're going to encounter our (infinitely precise) number, no matter how long we'd try.
(A physical voltmeter has a finite resolution, say, 0.10 V, so it's going to show you a predetermined voltage pretty often. But it would be a rounded approximation of the unpredictable real value.)