Pointwise Convergence For Induced Distributions

[Kreizhn]

Homework Statement

If $U \subseteq \mathbb R^n$ find a sequence of locally integrable functions $f_n \in L^1_{\text{loc}}(U)$ which converge pointwise, but whose induced distributions
$$\langle f_n, \cdot \rangle: C_c^\infty(U) \to \mathbb R, \qquad \langle f_n, \phi \rangle = \int_U f_n \phi$$
do not converge in the weak* topology.

The Attempt at a Solution

It would seem like the typical examples of "integral breaking" functions do not work here. In particular, I have tried examples of functions which spread ($\frac1n\chi_{(0,n)}$), congregate ($\frac n2 \chi_{[-1/n,1/n]}$) et cetera, but these do not seem to work. In particular, I think that it is because these choice of functions are not only locally integrable, but are integrable on all of $\mathbb R$ and converge weakly in $L^p$; in fact, each of the above converge to the Heaviside and delta distributions respectively. Any ideas would be useful.

 

[lippyka]

EDIT: Thanks to the help below I have made some headway. The idea is to set f_n(x) = \sum_{k=1}^{n} \frac{1}{2^k} \chi_{[x-1/2^k,x+1/2^k)} . That is, for each point we define a function which is 1 in a small neighborhood of that point and 0 elsewhere. Then, we can easily compute the integral of these functions as \int f_n = \sum_{k=1}^n 2^{-k+1} . This is not convergent, so we have found a sequence which does not converge weakly*. It is also easy to see that this sequence converges pointwise; in fact, it converges to the Dirac delta distribution in the pointwise sense.