Non-discrete locally compact Hausdorff groups which do not satisfy the second axiom of countability

Question

I think non-discrete locally compact Hausdorff groups which do not satisfy the second axiom of countability are pain in the ass. The major trouble is that the Haar measures on them are not necessarily $\sigma$-finite. I wish we could do math all right without them. Are there important examples of such groups which we cannot dispense with?

Answer 1

As noted in my comment, there is a large supply of compact, but not second countable groups, e.g. every uncountable product

$$ \prod_{i \in I} G_i, $$

where $I$ is uncountable and each $G_i$ is a compact group consisting of more than one element. These groups do not even fulfill the first axiom of accountability.

But from the formulation of your question, it is not entirely clear if you want to have examples. of relevant groups which are not second countable, or which are not $\sigma$-compact.

Here, you should observe that second countability implies $\sigma$-compactness, but not vice versa (as the example of the product of compact groups shows).