# Pointwise Convergence For Induced Distributions

1. Feb 5, 2012

### Kreizhn

1. The problem statement, all variables and given/known data

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.

3. 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.