Monotone increasing sequence of functions

Question

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 $\int f d \mu=\lim _{n \rightarrow \infty} \int f_{n} d \mu$.

Answer 1

Basically, monotone increasing functions mean a sequence of increasing functions. Your understanding is correct. There is nothing fancy. On a different note, I want to add something really helpful to solve a couple of exercises, which is: the conclusion of MCT also holds if

(1) we have a weaker version of pointwise convergence ie almost everywhere convergence.

(2) Please note that MCT does not hold for a decreasing sequence of functions. Can you find a counterexample?

(3) There is a version of MCT for decreasing sequence of functions. I want you to think at first, and it's easy.

(4) You can simply prove a simpler version of Tonelli's Theorem by MCT which is essentially the general linearity of integrals.

(5) My most favorite application of MCT is the Horizontal and Vertical Truncations (Please see Tao's book Page 101: An Introduction to Measure Theory).

I hope this helps. Good luck with the Analysis, and sorry for the digression.

Answer 2

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 $f_n \ge f_{n+1}$

Answer 3

When the term "monotone" is associated to the terms "increasing" or "decreasing", i.e. in monotone increasing or monotone decreasing, monotone then means the sequence need not be strictly increasing or strictly decreasing.

That is, the sequence can also be a constant sequence.

So

$f_n \le f_{n+1} \implies$ monotone increasing,

$f_n < f_{n+1} \implies$ (strictly) increasing.

There's nothing more to it, even in the case of MCT.