The breast band for exericse shown in the picture is called a strophium, fascia, fasciola, taenia, or mamillare. Its purpose was to hold the breasts and may also have been to compress them. The breast band was a normal, if optional, item in a woman's underwear. The bottom, loincloth-like piece is probably a subligar, but it was not a normal element of underwear, so far as we know.
Photo: Ancient Roman Women Exercising in Bikinis. Roman Mosaic From Villa Romana del Casale outside the town of Piazza Armerina, in Central Sicily. Mosaic may have been made in the 4th century A.D. by North African artists. CC Photo Flickr User liketearsintherain>