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 $$ X^{-1}X:=\{x^{-1}y\\,|\\,x\in X, y\in Y\} $$ is a neighborhood of the neutral element. Letting $X$ vary along all measurable subset 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.

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.