 
Summary: Probability spaces.
We are going to give a mathematical definition of probability space. We will first make some remarks
which will motivate this definition.
A fundamental notion in probability theory is that of an experiment. An experiment is an activity
which can be repeated any number of times, each repetition of which has an outcome. We require that
information about outcomes of past performances of the experiment provides no information
about future outcomes of performances of the experiment. The set of outcomes of the experiment
is called the sample space of the experiment. The points of the sample space, which are outcomes, are
sometimes called sample points. (It is possible that the experiment have only one outcome so that the
sample space has only one member, but this situation is probabilistically trivial.) Let us give two examples.
For the first example, suppose I take a coin out of my pocket, flip it three times and observe for each flip
whether it came up heads or tails. A natural sample space for this experiment would be the set of ordered
triples
{(H, H, H), (H, H, T), (H, T, H), (H, T, T), (T, H, H), (T, H, T), (T, T, H), (T, T, T)}.
Other sample spaces are possible. You may encode the outcomes as bit patterns, taking 1 for heads and 0
for tails, and ending up with
{(1, 1, 1), (1, 1, 0), (1, 0, 1), (1, 0, 0), (0, 1, 1), (0, 1, 0), (0, 0, 1), (0, 0, 0)},
or you could use the the numbers with the foregoing binary representations ending up with
{7, 6, 5, 4, 3, 2, 1, 0}.
But wait  maybe you don't care about the pattern of heads and tails. You might be interested in the sum
