Make your own exam questions --- follow up

I have received the following nice three exam questions (in two files):

• Problem 1 and 2
• Problem 3

They test measurable functions, convergence theorems and product measure. All of which is relevant.

Problems 2 in particular is very good. It gives a sequence of functions f_n that converges pointwise to a function f, but where

 lim_n \int f_n d\lambda \neq \int f d\lambda.

Having an understanding why you cannot use the MCT or the DCT here is good.