Let A be a subset of R, Lebesgue measurable and B a subset of R that is countable. I'm trying to show that the symmetric difference of A and B is Lebesgue measurable.
I've tried manipulating the definition of symmetric difference with the criteria from Lebesgue measurability but with no luck. Any direction or hints will be appreciated.
Every countable set is in the Lebesgue sigma algebra of R, therefore B is measurable. So $ A, A^c, B ^c$ are all in the sigma algebra, which is closed under countable unions and intersections.