2
$\begingroup$

I'm stuck trying to find an example of a sequence of functions $\{f_n\}_{n=1}^{\infty}$, which is nonnegative, such that $f_n \rightarrow 0$ uniformly and $\int\limits_0^\infty f_n=1$.

My first thought was a series of triangles of area 1 that move away from the origin that get taller and more narrow as they get further away.

  • 1
    You can’t afford to let them get taller: the resulting functions won’t approach $0$ uniformly. You need to do exactly the opposite: make them get shorter and wider, as Davide has done in his answer.2011-09-28

1 Answers 1

6

Put $f_n(x):=\begin{cases}\frac 1n&\mbox{ if }0\leq x\leq n\\\ 0 &\mbox{ otherwise}.\end{cases}$. Then $f_n$ is measurable since it's a constant times the characteristic function of a set of the Borel $\sigma$-algebra, nonnegative, $\displaystyle \int_{\mathbb R^+}f_n\lambda(dx)=1$ and $\displaystyle \sup_{\mathbb R}|f_n-0|=\frac 1n$, which shows the uniform convergence on $\mathbb R$ to $0$.