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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.

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  • $\begingroup$ If by "from below" you mean that $g_n$ is increasing monotonically to $0$, then $(ii)$ is in contradiction with the monotone convergence theorem. $\endgroup$ Jan 5 '21 at 14:45
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    – Ramiro
    Jan 6 '21 at 12:57
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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: $\lim_{n \to \infty} \int_{\mathbb{R}} g_n (x) dx = 0$.

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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 :

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

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