Are measurable automorphism of a locally compact group topological automorphisms?

The basic fact about locally compact groups is that you can recover the topology from the underlying measure space:

This is because, for any measurable subset $X\subset G$ of positive measure, the set $$ X^{-1}X:=\{x^{-1}y\mid x\in X, y\in X\} $$ is a neighborhood of the neutral element. Letting $X$ vary along all measurable subsets of positive measure you get a basis of neighborhoods of $e\in G$. By translating by group elements, you get a basis of neighborhoods of any element $g\in G$. And so you recover the topology on $G$.

Corollary:
Since the topology is entirely encoded in the measurable structure, an automorphism that respects the measurable structure, will also respect the topology, i.e., be continuous.

Here is a result by Adam Kleppner (Measurable homomorphisms of locally compact groups, Proc. Amer. Math. Soc., vol. 106, no. 2, 1989, 391-395): any measurable homomorphism between locally compact groups is continuous. Actually what he really needs, for a homomorphism $\alpha:G\rightarrow H$, is that $\alpha^{-1}(U)$ is measurable in $G$ for every open subset $U\subset H$.

You should look up "automatic continuity." Here is a paper by J.W. Lewin that may be of interest:

http://www.jstor.org/pss/2044356