Monotone increasing sequence of functions

measure-theorysequence-of-function

My understanding is a sequence of functions $(f_n)$ is monotone if it is increasing $f_n \le f_{n+1}$ or decreasing $f_n \ge f_{n+1}$.

What is a monotone increasing sequence of functions?

If it's just a increasing sequence of functions why say 'monotone increasing' and not just increasing?

so I think they have different meaning but I don't know which.

For further context this is about the Monotone Convergence Theorem in measure theory, that says,

Monotone Convergence Theorem

Let $(X, \mathbb{X}, \mu)$ be a measure space. Let $\left(f_{n}\right)$ be a monotone increasing sequence
of functions in $M^{+}(X, \mathbb{X}),$ which converges point wise to $f .$ Then $f \in$
$M^{+}(X, \mathbb{X}),$ and $\int f d \mu=\lim _{n \rightarrow \infty} \int f_{n} d \mu$.

Best Answer

When we say (monotone) increasing it implies that the sequence is monotone for that reason the term "monotone" can be omitted.

Usually we also distinguish

  • (monotone) strictly increasing when $f_n < f_{n+1}$
  • (monotone) increasing when $f_n \le f_{n+1}$

and

  • (monotone) strictly decreasing when $f_n > f_{n+1}$
  • (monotone) decreasing when $f_n \ge f_{n+1}$
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