I am trying to show that an open set in [0,1] is measurable, given that [0,x] is measurable set for each x in [0,1]. So I need to show (a,b) is measurable. Using the fact that measurable sets form a sigma algebra, I have managed to show that (a,b] is measurable. So (a,b+t] is measurable for any t>0. letting t ->0, can I then conclude that (a,b) is measurable? It seems a bit easy that I can just relax closed intervals to open intervals in this way.