A Brita filter will do nothing for treating water hardness.
As far as I know Brita filters contain a cation exchange resin, which will reduce Ca2+ concentration.
From the brita site (emphasis mine):
"How does a BRITA water filter cartridge work?
The majority of BRITA cartridges contain a combination of ion exchange resin and activated carbon. The carbon absorbs chlorine, pesticides and organic pollutants, improves taste and eliminates odours and discolouration. It also contains an inhibitor that prevents bacterial growth. The ion exchange resin reduces temporary hardness, which causes limecale. It also significantly reduces levels of metals such as copper and lead."