Integrability as prerequisite in the Monotone Convergence Theorem

Question

I know the Monotone Convergence Theorem formulated as follow:

Let $(f)_{n \geq 1}$ be an sequence of measurable functions isotonely converging to a function $f$ that is numerically measurable. If either $f_n \geq 0 \,\, \forall n\geq1 \,\, \mu-a.e.$ $\textbf{or}$ $f_1 \in \mathcal{L^1}(\mu)$ with $\mu$ being a measure, are fulfilled, the follwing equation holds:

$\\\\ \int f d\mu = \int \lim_{n \to \infty} f_n d\mu =\lim_{n \to \infty}\int f_n d\mu $

I am wondering why it's possible to dispense the prerequisite of non-negativity if we have integrability of the function $f_1$. Can anyone help out?

One of my friends guessed that both conditions enable monotonicity of integrals which is needed to proof the theorem and are therefore valid.

Thanks in advance.

Answer 1

Substract f1 on all sequence. Since f1 is L1 we have finite integral of f1