I recently tested a Sony Nex7 with Zeiss 24-70mm f2.8 stabilized lens, a Fuji X-T1 with Fuji 18-55mm f2.8-4.0 stabilized lens and the Olympus E-M1 with 12-40mm f2.8 lens. I have a bit of a tremor and besides still pictures I needed good video stabilization to capture my young granddaughter.
The Olympus is MUCH better, not even close, no contest. I purchased the Oly. BTW with the Oly you can shoot stills, while making a video. You skip frames in the process but may capture a precious moment at full resolution!