Requirements of Lebesgue measure theorems

Question

We are looking at the Lebesgue measure on $\mathbb{R}$ and ${(f_n)_{n\in \mathbb {N}}\colon \mathbb {R} \to \overline {\mathbb {R}}}$ with $f_n(x) = 1+\sum _{k=1}^n |x|^k$.

Does $(f_n)_{n\in \mathbb {N}}$ fulfill the requirements of

a. the dominated convergence theorem

b. Fatou's theorem

c. the monotone convergence theorem

?

I think that the answer to b is yes and to c no, is that correct? I'm not really sure about a.

Answer 1

For (b)&(c) the conditions are fulfilled, albeit trivially, since all of the integrals are infinite. None of these functions are integrable on $\mathbb{R}$, nor are they dominated by such, so the requirements of (a) are not fulfilled.