# condition for limit and differential

let $f:X\times [a,b] \rightarrow \mathbb R$. find conditions under which you may assert each of the following: $$\lim_{t \to t_0} \int_X f(x,t)d\mu = \int_X \lim_{t \to t_0} f(x,t)d\mu$$

$${d \over dt} \int_X f(x,t)d\mu = \int_X {\partial \over \partial t} f(x,t)d\mu$$

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These are standard facts in integration theory. Are you looking for conditions different than the usual ones? –  Siminore Nov 13 '12 at 15:51
i want to prove thats theorem in lebesgue integration, –  mshj Nov 21 '12 at 13:26