Fallen/rotten fruit on compost heap?

I have a large Bramley apple tree, with a lot of apples prematurely falling. Some apples rot already when on the tree; they turn brown and have little white spots or pustules that I assume may be full of fungal spores.Other apples that fall are green; they may have been infected by apple sawfly. My question is whether I can compost some or all of this; I wonder whether the ones with fungal spores on them are best thrown away.