I came across this expression but can't understand what: it means?
$P[X(s)=x_i]=P[{s_j:X(s_j)=x_i}]=P[{s_i}]$
I cannot understand what it means?
It is also used in the following result at the summation limits:
$P[X(s)=x_i]=P[{s_j:X(s_j)=x_i}]=\sum_{(j:X(s_j)=x_i)}P[{s_j}]$
These results are related to random variables.
They are the kind of mapping from the experimental universal set to the set of real nos on a real number line.
And these results are for assigning probabilities to the realizations of the random variable.
I added this so that anyone looking at this doesn't get confused as to where these expressions came from.
The colon is the “such that” colon from set builder notation.
Here, $P$ is a probability measure on some measurable space $S$. The notation $P[X(s) = x_i]$ (or I think more commonly and more cleanly, $P[X = x_i]$) is a short form for $$P(\{ s \in S : X(s) = x_i \})$$ where $E := \{ s \in S : X(s) = x_i \}$ is now a (measurable) subset of $S$, i.e. an event, so we can compute its probability $P(E)$. The formula $P[s_j : X(s_j) = x_i]$ is supposed to denote the same thing. (Leaving off the set braces for brevity.) The last version, $P[s_i]$, is not generally equal to the other two; this must use the definition of $X$ and $x_i$ somewhere.
(By the way, we use the first short form because in general one wants to think about the underlying probability space $S$ as little as possible; it’s basically an “implementation detail”.)
In the second part of your question (concerning the summation), the colon plays a similar role: The sum is over all indices $j$ such that $X(s_j) = x_i$. This notation avoids having to define the set $J = \{ j : X(s_j) = x_i \}$ only to write $\sum_{j \in J} P[s_j]$.