I have some trouble with a theoretic-like exercise about measure theory, and $\pi$- and $\lambda$-systems, and I would like to have some help. The problem is stated in the book Mathematical Statistics, Jun Shao, exercise 5 of section 1.6, page 74:
I am somewhat puzzled because I expected $\mathcal{D}$ to be a $\sigma$-algebra (the exercise would be vacuosly easy then), but to be a $\lambda$-system is strictly weaker. I suppose one has to use some trick about $\pi$-systems, but I cannot guess what to use because of the non-constructive definition of the $\sigma$-algebra generated by some subset.
Any suggestion is welcome. Thanks in advance.