Now we can define what it means for a general real-valued function (not just a simple function) to be integrable: a function is integrable if there is a mean Cauchy sequence of integrable simple functions which converges in measure to . We then define the integral of to be the limit
But how do we know that this doesn’t depend on the sequence ?
We recall that we defined
which must be measurable for any measurable function . This is the only part of the space that matters when it comes to integrating ; clearly we can see that
since is zero everywhere outside .
Now, if both and converge in measure to , then we can define to be the (countable) union of all the and . Just as clearly, we can see that
where is the indefinite integral of , and is the indefinite integral of . Then if we use to define the integral of we get
while if we use we get
But we know that since and both converge in measure to the same function, the limiting set functions and coincide, and thus . The value of the integral, then, doesn’t depend on the sequence of integrable simple functions!