Monotone Convergence Theorem for non-negative decreasing sequence of measurable functions

Solution 1:

The problem is that $-f_n$ increases to $-f$ which is not non-negative, so we can't apply directly to $-f_n$ the monotone convergence theorem. But if we take $g_n:=f_1-f_n$, then $\{g_n\}$ is an increasing sequence of non-negative measurable functions, which converges pointwise to $f_1-f$. Monotone convergence theorem yields: $$\lim_{n\to +\infty}\int_X (f_1-f_n)d\mu=\int_X\lim_{n\to +\infty} (f_1-f_n)d\mu=\int_X f_1d\mu-\int_X fd\mu$$ so $\lim_{n\to +\infty}\int_X f_nd\mu=\int_X fd\mu$.

Note that the fact that there is an integrable function in the sequence is primordial, indeed, if you take $X$ the real line, $\mathcal M$ its Borel $\sigma$-algebra and $\mu$ the Lebesgue measure, and $f_n(x)=\begin{cases} 1&\mbox{ if }x\geq n\\\ 0&\mbox{ otherwise} \end{cases}$ the sequence $f_n $ decreases to $0$ but $\int_{\mathbb R}f_nd\lambda =+\infty$ for all $n$.