I have been looking at Kolmogorov's book "Introductory Real Analysis" and have become stuck at the problem 4a) on page 301.
In this problem we are given $f$ a nonnegative and integrable function on $A$, a set of finite measure. We are asked to show that
$\int_A f(x) d \mu \ge 0$
However, how would we prove this? In this text the Lebesgue integral of a function $f$ is defined to be (when the limit exists);
$\int_A f(x) d\mu = \lim\limits_{n \rightarrow \infty} \int_{A} f_n(x) d\mu$
where $f_n$ is a sequence of integrable simple functions that converges uniformly to $f$.
I have attempted to show this by showing that the sequence of simple functions converging to a nonnegative function, is a sequence of nonnegative functions. However this isn't true, take $f_n = -1/n$ and $f = 0$. How else could I go about this?
Thanks.