The brassiere has been around in one form or another at least since Greek women wore a chest binding called a mastodeton or apodesmos while exercising. When to date the invention of the modern brassiere to presents a bit of a problem -- it depends on how one defines the contraption.

Metallic constructions that looked like bras and did serve to hold matters rigidly in place were around as early as 1859, as a patent granted to Henry S. Lesher of Brooklyn, NY, shows. However, to think of this monstrosity as a garment would be to equate medieval armor with a leisure suit.
Closer to the concept was Clara P. Clark's 1874 "improved corset," a fabric construction that bore a resemblance to today's long-line bra. The fundamental element of her design was the pattern for a fabric breast pocket system held up by shoulder straps that crisscrossed the back.
A French woman named Herminie Cadolle was the genius behind the cups and straps. Cadolle came up with the concept in 1889, originally dubb ...more

Source: Yahoo Answers

The one who introduced could be a BRA hmin.

According to Life magazine, in 1889 Herminie Cadolle of France invented the first modern bra.[42] It appeared in a corset catalogue as a two-piece undergarment, which she originally called the corselet gorge, and later le bien-tre (or "the well-being"). Her garment effectively cut the traditional corset in two. The lower part was a corset for the waist, the upper supporting the breasts by means of shoulder straps. Her description reads "designed to sustain the bosom and supported by the shoulders". She patented her invention and showed it at the Great Exhibition of 1889.

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Mary Phelps Jacob