I need to compute the integral $$\int _0 ^ \infty xe^{-kx} dx, \,k\geq1$$ by means of measure theory. Integration by parts will not work here. Also I used the Taylor series of the exponential function with the idea in mind to use the fact that for positive functions one can interchange sum symbol and integral, but it did not lead me further.
I will appreciate your help. Thanks.
Hint: $$\int_0^\infty x e^{-kx}\; dx = - \frac{d}{dk} \int_0^\infty e^{-kx}\; dx $$ Use "measure theoretical means" to justify the interchange of derivative and integral.