Does any locally compact topological group which is not Hausdorff have a Haar measure?

Question

In 'Linear Analysis and Representation Theory' by Steven Gaal at the end of Chapter IV, page 227, the author claims that any locally compact topological group $G$ which is not Hausdorff has a Haar measure and I question the validity of this claim.

At the end of his argument he has constructed an invariant positive functional on a subspace of $C_0(G)$ but gives no proof that this functional can be extended to the full $C_0(G)$ whilst maintaining the positivity and invariance. He also doesn't mention any version of the Riesz Representation Theorem for non-Hausdorff spaces (I am also not convinced by his argument that a continuous function $f:G \to [0,1]$ with compact support and $f \equiv 1$ on $C$ exist if $C$ is compact but not necessarily closed).

So my question is: Can the gaps in these arguments be filled and so does $G$ indeed have a Haar measure or does there exist a counterexample?

See also this question where in the comments some serious problems were pointed out (after 7 years) with Simon Rubinstein-Salzedo's ON THE EXISTENCE AND UNIQUENESS OF INVARIANT MEASURES ON LOCALLY COMPACT GROUPS.

Thank you!

Answer 1

Let $G$ be a topological group.

(1) $G$ is Hausdorff iff $\{1\}$ is closed. One direction is clear, and conversely if $\{1\}$ is closed, then its inverse image under the map $(g,h)\mapsto gh^{-1}$ is the diagonal of $G\times G$ which is then closed, and hence $G$ is Hausdorff.

(2) In general, let $N$ be the closure of $\{1\}$. Then $G/N$ is Hausdorff, by (1).

(3) Every open subset of $G$ is $N$-invariant. Hence, the Borel $\sigma$-algebra of $G$ consists exactly of the inverse image of the Borel $\sigma$-algebra of $G/N$.

(4) In particular, the Borel $\sigma$-algebra of $G/N$ and $G$ are canonically isomorphic (through taking the inverse image), and this induces a canonical identification "pullback" from the set of measures on $G/N$ and those on $G$, which restricts to an identification between Radon measures on $G/N$ and on $G$, and also restricts to an identification between left-invariant measures on $G/N$ and $G$, and in turn, between Haar measures (defined as Radon left-invariant measures). Note that in the definition of Radon (locally finite + inner regular), one needs to use "closed compact" in the definition of inner regular).

(5) Define "compact" in the usual way (without "Hausdorff") and "locally compact" as "every point has a closed compact neighborhood". Then the closed compact subsets of $G$ are exactly the inverse images of the closed compact subsets of $G/N$. Then $G$ is locally compact iff $G/N$ is locally compact. In this case, the pullback of a Haar measure on $G/N$ defines a Haar measure on $G$.