My textbook defined M.C.T. by
for $\{f_k\}$ be a sequence of measurable functions on $E\subset\mathbb{R}^n$,
- If $f_k\nearrow{}f~~a.e.$ on $E$ and there exists $\phi\in{}L(E)$ such that $f_k\ge\phi~~a.e.$ on $E$ for all $k$, then $$\int_E{}f_k\to\int_E{}f$$
- If $f_k\searrow{}f~~a.e.$ on $E$ and there exists $\phi\in{}L(E)$ such that $f_k\le\phi~~a.e.$ on $E$ for all $k$, then $$\int_E{}f_k\to\int_E{}f$$
In the M.C.T. in the sense of the sequence of real numbers, when a sequence is increasing and bounded above, then its supremum is the limit. Conversely, when a sequence is decreasing and bounded below, then its infimum is the limit.
I do not understand why $f_k$ is bounded below by $\phi$ although $f_k$ are monotonically increasing functions, i.e. $f_k\le{}f_{k+1}$. Conversely, I do not understand why $f_k$ is bounded above by $\phi$ although $f_k$ are monotonically decreasing functions, i.e. $f_k\ge{}f_{k+1}$.