Michael:
For a 1-1/8" shelf, I would dado 1/4" x 3/4" steel bar into the front edge full width, with the 3/4" face vertical. Epoxy the steel in place and trap it behind additional glued in wood blocking that is flushed up, so that when you add your nosing, it will have a full 1-1/8" wood glue face. If you go this route, you may want to change your shelf thickness to 1-1/4", and face glue 3/4" and 1/2' material together. Dado the back edge of the shelf into a 3/4" cabinet back as well, as Rich suggests, and you have done what you can do without a change of materials or design.
Overkill, maybe, but better safe than sorry. If the shelf is dadoed into the sides and back, you should be able to stand on that shelf with minimal flex. I once did a bookcase job for couple that were both NASA scientists, where the interior designer told me not to worry about structure, because the bookcase shelves would only hold knick-knacks, only to return to the job to find the entire cubic area between shelves packed solid with books, with additional books on the floor. I was lucky only insofar as the packing of books in the entire bookcase helped to keep the shelves from sagging too much, but I can't imagine it was easy to pull out a book from the bottom shelf.
That's when I learned to ignore designers. Hope this helps. Good luck
TonyF