The APS-C sensor size has some advantage over Micro Four Thirds for background blur. But that does not mean M4/3 is incapable of blur! For this trial I will use a Tamron 17-50mm f/2.8 lens on an APS-C Canon 50D, vs an Olympus 45mm f/1.8 prime on a M4/3 Olympus E-M5.
First case is both lenses at f/2.8. The background blur is about the same. Why, when the Canon sensor is a little bigger? Because the distances are all the same and the focal lengths are nearly the same.
Next case is both lenses wide open. F/1.8 for the Olympus gives stronger blur.
Final case is the often cited “equivalence” situation. Where the larger sensor is set to a smaller aperture.
But the Canon shot shows weaker blur. So – when the smaller sensor camera uses a tighter frame (allowing the same focal length) – the equivalence game becomes null and void.