Good morning Math Stack Exchange
In my journey to learn Measure Theory, I wish to understand with clarity the link between a random variable's distribution function (df) and it's distribution. I believe some would call the distribution a density but in Resnick's "A Probability Path", the book I'm using, he calls it distribution.
I've been reading the chapter on Integration and Expectation and oddly enough, there isn't a theorem on how we can get the distribution from its df. Undergraduate classes would be quick to say "differentiate the df to get the distribution". But I'm sure there are numerous df which are not continuous hence not differentiable, i.e., $F(x)=\sum\frac{1}{2^n}1_{x\le n}$. For such df's, is it possible, and under what conditions, for us to get the df?
To me, I feel that we can easily get the df from the measure on defined on $(\Omega,\mathcal{F})$ namely $F(x)=P[X
Thank you!