I was reading the wikipedia page on convolutions earlier and stumbled upon the section "Convolutions on groups", which offered the following definition:

If $G$ is a suitable group endowed with a measure $\lambda$, and if $f$ and $g$ are real or complex valued integrable functions on $G$, then we can define their convolution by $$(f*g)(x)=\int_Gf(y)g(y^{-1}x)d\lambda(y)$$

I'm still pretty new to group theory, so I've never seen an integral defined over a group before, and I haven't been able to figure out what they mean by $G$ being "endowed with a measure $\lambda$".

What do they mean by a measure on the group, and how does integration over a group work?

For reference, the wikipedia page in question is here: https://en.wikipedia.org/wiki/Convolution#Convolutions_on_groups

  • $\begingroup$ Read a book about measure theory and the lebesgue integral. $\endgroup$ – Paul K Nov 13 '16 at 8:44

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