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.

• If by "from below" you mean that $g_n$ is increasing monotonically to $0$, then $(ii)$ is in contradiction with the monotone convergence theorem. Jan 5 '21 at 14:45

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: $$\lim_{n \to \infty} \int_{\mathbb{R}} g_n (x) dx = 0$$.

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 :

$$g_n\leq 0, g_n\text{ converges pointwise to 0, and : }\lim_{n\to\infty}\int_\Omega g_nd\mu=-1$$