for the Fisher-Information to be defined certain regularity conditions have to be fulfilled (like in Lemma 5.3. in Theory of Point Estimation by E.L. Lehmann or on slide 2 here: http://www.stat.nus.edu.sg/~stachenz/40-530Statistics-6.pdf).
I know that any exponential family satisifies these conditions. But how does one formally show property (iv) on slide 2 in the last link for exponential families, i.e. that one can interchange derivative and integral/expectation? Should one use the dominated convergence theorem? And how does one prove the second assumption (ii) on slide 5 of the link? If you know literature containing the proof I am interested in reading it.
Looking forward to your answers!
Thanks!