I have following sequence:
$$ f_n = -\frac{1}{n} $$
I wanted to show, that it converges to $f = 0$, but my book says, that it doesn't, because the condition
$$ \int_{\mathbb R} f_1 d\lambda > -\infty $$
is not met.
My question is - How can I compute Lebesgue integral of $f_1$ ?
$$ f_1 = - \frac{1}{1} = -1 $$
And the integral is:
$$ \int_{\mathbb R} -1 d\lambda $$
I know, that the integral is defined like this:
$$ \int_E f d\lambda = sup \left\{ \int_E s d\lambda: 0 \leq s \leq f \right\}, $$
where $s$ is considered a simple function, so its integral is defined like this:
$$ \int_E s d\lambda = \sum^k_{i=1} \alpha_i \lambda(A_i \cap E), $$
where $A_i = \{x \in \mathbb R^n: s(x) = \alpha_i \in \mathbb R^+_0\}$.
But still, unfortunately, I'm not able to "connect" all those definitions and compute the integral...