Suppose that ๐พK K is a kernel from (๐,๐ฎ)(S,S) (S, \mathscr S) to (๐,๐ฏ)
Shouldn't it be a Kernel on a single measure space in order to talk about invariance?
For instance, \( \mu K \) is a measure in \( (T, \mathcal{T}) \), not a measure in \((S, \mathcal{S})\)
Same would happen for item 2, \(. Kf \) is a function in S, rather than a function in T.