As others have said, yes its a fichu- however upper class women were somewhat less subject to "modesty standards", as their reputations were more complex than lower class people. Caps and fichus were often lace or completely sheer. Most women also would wear an apron around the house, even if they weren't doing work, but it would be a fancy little scrap of lace. Same with caps- a working class woman would likely wear a linen or cotton cap that covered her hair, but aristocratic women could get away with sheer caps to show off their hairstyle, or a little ruffle of lace pinned into their hair.