Let f and g be measurable functions from X to the extended reals. Prove that the sets

A={x:f(x)<g(x)}

and

B={x:f(x)=g(x)}

are measurable.

And then prove that the set of points at which a sequence of measurable real valued functions converges is measurable.

To show that A is measurable I am trying to work with intersections. I know that f is measurable iff {(a,inf]} is measurable for all a in R. Similarly for g. Can I simply take the intersection of such sets to get A? Is the intersection of measurable sets measurable?

Can I do something similar for B?

Assuming A and B are measurable, I suppose I have to use that to show the second part of the question, but I dont see how.