# Lebesgue measure theorems – Mathematics Stack Exchange

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

a. Does $$(f_n)_{nin mathbb {N}}$$
fulfill the requirements of the dominated convergence theorem?

b. Does $$(f_n)_{nin mathbb {N}}$$
fulfill the requirements of Fatou’s theorem?

c. Does $$(f_n)_{nin mathbb {N}}$$
fulfill the requirements of the monotone convergence theorem?

I think that b is true and that c is wrong, is that correct? I’m not really sure about a.