Buying fresh fruit and vegetables at the market is good for two reasons - the Epicurean doesn't really have a good selection of those, and both that and the other supermarkets (some of which have a better choice) can be unbelievably expensive - because much of their fresh produce isn't produced locally - it comes on the weekly Tropical Lines boat from Florida. You may find that at busy times there is a better selection of anything fresh in supermarkets on (if I remember correctly), Tuesday and Wednesday - the Jolly Harbour Epicurean can get thin on fresh fruit juices and fruit & veg on Sunday/Monday. I believe the government of Antigua is keen to support more local grown produce, but that isn't yet evident in the bigger supermarkets.
By the way, check all your prices when shopping - prices of simple basics like pasta and bottled pasta sauce, pesto, etc, can vary by 300% or more, depending on whether the product is US produced or imported from Eurpoe. I'm vegetarian, but I've read here that meat and fish can be good value if you know where to get it.