Example of a strictly positive bound function with constant norm that in mean converges to zero

Question

It is well known that $\int_{0}^{1}\sin(nx)dx\rightarrow 0$ but $\int_{0}^{1}\sin^{2}(nx)dx$ converges to a positive constant.

Is there a sequence of strictly positive, uniformly bounded functions with the same property? That is $\int_{0}^{1} f^{n}(x)dx \rightarrow 0$ but $\int_{0}^{1} f^{{n}}(x)^{2}dx$ converges to a strictly positive constant.

I tried a sequence of Pareto distributions, but they become unbounded. The functions $\left|\sin(nx)\right|$ also converge to a constant mean.

Any help would be appreciated.

Try $f_n(t)=\sqrt{n}1_{[0,1/n]}(x)+\frac{1}{n}1_{]1/n,1]}(x)$. — Kelenner, Jun 06 '17 at 14:06
Do you mean that $f_n$ is uniformly bounded, $f_n(x)<K$ for all $x,n$? — Chappers, Jun 06 '17 at 14:08
Yes, sorry I should clarify, uniformly bounded functions. I'll edit my question. — Akshay Shanker, Jun 06 '17 at 14:21
If they're uniformly bounded, say by $M$, then $f^n(x)^2 \leqslant M\cdot f^n(x)$. — Daniel Fischer, Jun 06 '17 at 14:22
Ok that answers my question, very obvious, I should have seen that. Thank you! — Akshay Shanker, Jun 06 '17 at 14:23

Akshay Shanker · Answer 1 · 2017-06-15T06:21:54.660

For the sake of completeness, I will answer my own question.

Daniel Fischer points out that if a sequence of postive $f_n$ on $[0,1]$ is uniformly bounded point wise by a constant M, then $f_{n}^2 \leq Mf_{n}$. Let the mean of $f_n$ converge to zero, we have:

$$ \int_{0}^{1} f_{n}(x)^{2}dx \leq M\int_{0}^{1} f_{n}(x)dx $$

Thus the square integral of the family of functions $f_n$ converges to zero as the RHS above converges to zero.

Example of a strictly positive bound function with constant norm that in mean converges to zero

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