I would like to start studying Measure Theory in the summer of 2024. Let me tell you a bit about myself:
I'm a graduate student in Mathematics from Brazil. I suspect that, by luck, I ended up being approved for a Master's in Mathematics at an excellent graduate program. My university (where I'm graduating) is far from being one of the best and I don't have a good grounding in mathematical analysis. I've recently dedicated myself a lot to studying real analysis, as you can see in some of my recent posts, and I managed to get approved for the master's degree (they're different universities).
Looking at the class notes from last summer at the university where I'll be doing my master's, I had a hard time understanding some of the results. For example:
Theorem 1. Let $(f_n)_n$ be a sequence of $\mathcal{M}(X, \mathcal{X})$. Then the functions defined by: $f(x) = \text{inf}_{n} \ f_n(x)$, $F(x) = \text{sup}_{n} \ f_n(x)$, $f^*(x) = \text{sup}_{n \geq 1} \{ \text{inf}_{m \geq n} \ f_m(x)\}$ and $F^*(x) = \text{inf}_{n \geq 1} \{ \text{sup}_{m \geq n} \ f_m(x)\}$ are measurable functions.
Theorem 2. If $f,g \in \mathcal{M}(X, \mathcal{X}),$ then $fg \in \mathcal{M}(X, \mathcal{X})$.
Lema 3. If $f \in \mathcal{M}(X, \mathcal{X})$ is non-negative then there exists a sequence $(\phi_n)_n$ in $\mathcal{M}(X, \mathcal{X})$ such that:
- $0 \leq \phi_n(x) \leq \phi_{n + 1}(x)$, for all $x \in X$ and for all $n \in \mathbb{N}$.
- $f(x) = \lim\limits_{n \longrightarrow \infty} \phi_n(x)$, for all $x \in X$.
- Every $\phi_n(x)$ takes on a finite number of real values.
So my question is: Can I start the study of Measure Theory from the definition of Measure without getting too bogged down?
I have a reasonable understanding of what a sigma-algebra is, what a measure space is and, from the definition of measure, the subject seems more understandable to me. I also plan to ask the teacher about these three theorems next year.