Can one define an uncountable infinite product?

Question

The continuous version of a sum is commonly called an integral. But what would be the continuous version of a product? The same question in "pictures": $$ \sum \rightarrow \int $$ $$ \prod \rightarrow ?? $$ I thought that maybe one could argue as follows: $$ \prod_n a_n = e^{\ln \prod_n a_n} = e^{\sum_n \ln a_n} \rightarrow e^{\int \ln a(n) dn} $$ So the continuous version of a product could be defined like $$ \prod \rightarrow e^{\int \ln} $$ Does this make sense?