suppose we have a space , where is Lebesgue measure. and let be a sequence of functions:

Show in measure and almost eveywhere.

To show the convergence in measure, for given , let . Then e^{-x})^n > \epsilon \}=" alt="E=\{x:f_n>\epsilon \}=\{xe^{-x})^n > \epsilon \}=" /> . So as . So in measure.

here is my question. In this question, i dont really see the difference between the convergence in measure and convergence almost everywhere. Can i prove the convergence almost everywhere in the same way i did above? or i have to show the different proof?