I am teaching myself measure theory and I working through http://homepages.uconn.edu/~rib02005/real.html. In exercise 7.3, they ask:
Give an example of a sequence of non-negative functions $f_n$ tending to $0$ pointwise such that $\int f_n→0$, but there is no integrable function $g$ such that $f_n ≤ g$ for all $n$.
I am a novice in pure mathematics, so I would like some help in determining whether or not I am on the right path.
First, I believe I must find a sequence of function for which the $\lim_{n\rightarrow \infty} f_n = 0$ and $\int f = 0$ if I interpret the question correctly.
I am approaching this problem by exploiting the fact that $f_n \leq g$ must hold for all $n$. Therefore, I only need to find a single $f_n$ for which the inequality is violated, as long as the function and its integral are zero in the limit.
I initially thought of $f_n=|\frac1n|$, since this is a non-negative function that satisfies the constraints above. However, I am not completely convinced if my reasoning is sound, or that I have understood the question.
Can you help me improve, or correct, my answer?
The problem with your suggestion is that $\int f_n$ does not seem to be finite if integrating a positive constant along the whole real line.
You could try something like
$$f_n(x) = \begin{cases} n &\mbox{if } 0 \lt x \lt \frac1{n^2} \\ 0 & \mbox{otherwise. } \end{cases}$$