Do there exist examples of integrable functions with these properties? Question:

Do there exist examples of sequences of integrable functions $(f_n)_n$
and $(g_n)_n$ on $\mathbb{R}$ (with it standard measure structure)
converging pointwise to zero and satisfying:
(i) $\lim_{n \to \infty} \int_{\mathbb{R}} f_n (x) dx = + \infty$
(ii) $g_n \leq 0$, $(g_n)$ converging to zero (from below) and
$\lim_{n \to \infty} \int_{\mathbb{R}} g_n (x) dx = -1. $
If so, provide an example; if not, explain why not.

Attempt: For (i), I think yes. I consider the standard measure space $(\mathbb{R}, \beta, \lambda)$ where $\lambda$ is the Lebesgue measure on the Borel $\sigma$-algebra $\beta$. I let $f_n(x) = \frac{1}{n} \chi_{[0, n^2]} (x)$ where $\chi$ is the indicator function. Then $|f_n(x)| \leq 1/n$ for all $x \in \mathbb{R}$, and so $f_n \to 0$ pointwise (even uniformly), but $$ \lim_{n} \int_{\mathbb{R}} f_n(x) dx = \lim_n \frac{1}{n} n^2 = + \infty. $$
For (ii), I think this is impossible because it contradicts the Dominated Convergence Theorem. In class, we saw this theorem as:
Theorem:

Let $(\Omega, \mathfrak{M}, \mu)$ be a measure space and let $(f_n)_n$
be a sequence in $\mathcal{L}^1 (\Omega, \mathfrak{M}, \mu)$
converging pointwise to a function $f: \Omega \rightarrow C$. Suppose
there exists a positive integrable function $g: \Omega \rightarrow [0,
 + \infty]$ such that $|f_n| \leq g$ for all $n \in \mathbb{N}$. Then $f \in \mathcal{L}^1 (\Omega, \mathfrak{M}, \mu)$ and $$ \lim_n
 \int_{\Omega} | f_n - f| d \mu = 0$$ and $\lim_n \int_{\Omega} f_n d
 \mu = \int_{\Omega} f d\mu. $

So now I was thinking that (ii) would contradict this theorem. If $g_n \leq 0$, then $(g_n)_n$ is dominated by the zero function. So we should have $\lim_n \int_{\mathbb{R}} g_n dx = 0. $ Is this reasoning correct?
Thanks for any help.
 A: This reasonning isn't correct because the there is an absolute value in the domination, so $ g_n\leq 0$ turns into $|g_n|\geq 0 $.
And if you take $g_n=-\frac{1}{n}\chi_{[0,n]} $, you have :
$$g_n\leq 0, g_n\text{ converges pointwise to 0, and : }\lim_{n\to\infty}\int_\Omega g_nd\mu=-1 $$
A: Here are two simpler examples:
Consider $f_n= \chi_{[n,n^2]}$    and  $g_n=-\chi_{[n,n+1]}$.
They are sequences of integrable functions $(f_n)_n$
and $(g_n)_n$ on $\mathbb{R}$
converging pointwise to zero and satisfying:
(i) $\lim_{n \to \infty} \int_{\mathbb{R}} f_n (x) dx = + \infty$
(ii) $g_n \leq 0$, $(g_n)$ converging to zero (from below, but not monotonically) and
$\lim_{n \to \infty} \int_{\mathbb{R}} g_n (x) dx = -1. $ In fact $ \int_{\mathbb{R}} g_n (x) dx = -1$, for all $n$.
Remark 1: Note that $g_n$ is not dominated by $0$. To be dominated by $0$ means $|g_n|\leqslant 0$, which is not the case.
Remark 2:If you require the $g_n$ converges to zero (from below) monotonically, then (ii) would be impossible.  For all $n$, we would have $|g_n|\leqslant |g_1|$ and since $g_1$ is integrable, so is $|g_1|$ and then by the Dominated Convergence Theorem, we would have:
$\lim_{n \to \infty} \int_{\mathbb{R}} g_n (x) dx = 0$.
