I've just come across the term product integral whilst reading a survival analysis paper. I'm surprised that it's the first time I've seen this but it seems that the idea went out of fashion and has only really been promoted in survival analysis circles because of the usefulness linking cumulative hazards and Kaplan-Meier estimates.
The idea is simple, especially when you know what regular, run-of-the-mill integration is. Where (sum) integration is the asymptotic limit of the sum of smaller and smaller intevals beneath a curve i.e. the continuous analogue of a summation, then the product integral is the product of smaller and smaller powers so these values approach 1 from above i.e. this is the analogue of taking products instead of sums.
The current notation I've seen a lot of was proposed by Gill and Johanson. I found this article by Gill useful in explaining what's going on.
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