I am not specialist on Topological Group Theory, I apologize if this is a trivial question.
Question. If $G_1=G_2$ are amenable topological groups what additional hypothesis we have to consider on the group, in order to prove that $G_1\times G_2$ is amenable ?
Following Leinster, in this question Why are abelian groups amenable?,
"The direct product of two amenable groups is amenable. This isn't exactly trivial, but the measure on the product is at least constructed canonically from the two given measures."
So discreteness of the $G_1$ and $G_2$ are enough to prove that $G_1\times G_2$ is amenable and also we do not need to suppose that $G_1=G_2$.
Looking for a proof, in more general cases, I found the following statement:
"... direct product $G_1\times G_2$ of two countable amenable groups can not be amenable."
in the paper, On Subadditive Processes on Direct Product of Countable Amenable Groups by Seyit Temir - Publications De l'Institute Mathématique (2002), 119-122.
Since the author did not mention if the example needs two different groups and I have no access to the paper containing this information, I decided to post this question.
I would be grateful if you could point me out some references discussing about this problem.