I'm facing now for the first time the topic in the title, and found myself having hard times to figure out what a jump function is. If you look for anything on google you'll always find stuff about C++ or assembly jumps which is of course not what I'm looking for.
Here's what I have in my notes:
Suppose being assigned $n$ real numbers $t_1,t_2,\dots,t_n$ where $\forall i, \;\;t_i\in\mathbb{R}$, and the same amount of corresponding jump values $h_1,h_2,\dots,h_n$. Then we define the jump function as $F(t):\mathbb{R}\to\mathbb{R}$ given by $F(t) = \sum\limits_{j\;|\;t_j \leq t} h_j$
and this is a probability measure if $F(t)$ has $1$ as horizontal asymptote.
This is quoted as example in my notes for a Lebesgue-Stieltjes measure, but still I'm not getting how is this related, and moreover I'm not getting how the plot of a jump function should look like (maybe like steps? It resembles a cumulative distribution function for a discrete random variable).
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$\begingroup$The name "jump function" is really unusual. You will find lots of information if you use the common name step function.
Your function $F$ takes the values $0,h_1,h_1+h_2,\ldots,h_1+\cdots+h_n$. If we name $k_j=\sum_{\ell=1}^jh_\ell$, we can write $F$ in the more common way $$ F(t)=\sum_{j=1}^n k_j\,1_{[t_j,t_{j+1})}. $$ And yes, it looks like stairs, since $k_1<k_2<k_3<\cdots<k_n$.
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