E,
I am working on some homework problems in which we are being asked to evaluate integrals using measure integration.
An example problem I am working on is to evaluate:
$$ \int_{[0,1]} f\,dm$$ Where $f(x) = x$ for every $x \in \mathbb R$ and $m$ is Lebesgue Measure.
I am posting here because I am struggling to understand how exactly to evaluate a measure integral and what the answer should even look like.
Firstly, I would like to explain what I understand and what we have covered so far in lectures. So far in our latest class we proved The Dominated Convergence Theorem and have of course shown recently the key properties of Lebesgue integrals along with proving The Monotone Convergence Theorem and Fatou's lemma.
I understand that we often write out simple functions in their canonical representation and can evaluate integrals that way however I have never seen a worked example. I also understand the following definition presented in class:
Let $f:X\to [0,\infty]$ be measurable then define: $\int_X fd\mu := \sup\{\int_X \phi d\mu \mid\phi: X \to [0,\infty)$ is simple & measurable with $0 \le \phi \le f\}$
What I do not understand and is causing me much confusion is how to apply the definitions and theorems to actually solve an integral.
I would greatly appreciate an explanation or some suggestions on how to evaluate such an integral and how to better understand these types of exercises rather than just an answer if possible. I would like to fill in the gaps in my understanding that is preventing me from knowing how to answer these type of questions despite knowing and having some understanding of the course material.
Thanks!
P.S: Please do not hesitate to suggest any formatting or editing corrections I have no experience with MathJax.
Since you know dominant convergence theorem, we can make use of it.
Define, for $i=1,2,..,n$, $$f_n(x) = \frac{i}{n} \quad \text{when } x\in [\frac{i-1}{n},\frac{i}{n})$$ Then $x\leq f_n(x)$ for all $x\in [0,1)$. Note that $f_n(x) $ converge pointwise to $x$. Dominant convergence theorem says: $$\int_{[0,1)} x dx = \lim_{n\to\infty}\int_{[0,1)} f_n(x) dx$$ $f_n(x)$ is a step function, which makes the limit easy to calculate. We have $$\int_{[0,1]} f_n(x) dx = \frac{1}{n}\left( \frac{1}{n} + \frac{2}{n} +... + \frac{n}{n} \right) = \frac{1}{n}\frac{n(n+1)}{2n}$$