(a) Consider the topology $\mathcal{T}$ on the natural numbers $\mathbb{N} \subset \mathbb{R}$ induced by the standard topology on $\mathbb{R}$. Prove it is the discrete topology; i.e. $\mathcal{T}=\mathcal{P}(\mathbb{N})$ is the power set of $\mathbb{N}$.

(b) Describe the corresponding Borel sets on $\mathbb{N}$ and prove that any function $f: \mathbb{N} \rightarrow \mathbb{R}$ or $f: \mathbb{N} \rightarrow[0,+\infty]$ is measurable.

(c) Using Lebesgue integration theory, define $\sum_{n \geqslant 1} f(n) \in[0,+\infty]$ for a function $f: \mathbb{N} \rightarrow[0,+\infty]$ and then $\sum_{n \geqslant 1} f(n) \in \mathbb{C}$ for $f: \mathbb{N} \rightarrow \mathbb{C}$. State any condition needed for the sum of the latter series to be defined. What is a simple function in this setting, and which simple functions have finite sum?

(d) State and prove the Beppo Levi theorem (also known as the monotone convergence theorem).

(e) Consider $f: \mathbb{R} \times \mathbb{N} \rightarrow[0,+\infty]$ such that for any $n \in \mathbb{N}$, the function $t \mapsto f(t, n)$ is non-decreasing. Prove that

$$\lim _{t \rightarrow \infty} \sum_{n \geqslant 1} f(t, n)=\sum_{n \geqslant 1} \lim _{t \rightarrow \infty} f(t, n) .$$

Show that this need not be the case if we drop the hypothesis that $t \mapsto f(t, n)$ is nondecreasing, even if all the relevant limits exist.