# Uniform Limit of Integrable Functions with Non-Compact Domain is Not Necessarily Integrable.

I am trying to find an example of a sequence of functions $(f_n)$ on $\mathbb{R}$ such that: Each $f_n$ is integrable, $f_n\rightarrow f$ uniformly; the limit function $f$ is not integrable. Integrable means here normal Riemann integrable. Any help will be appreciated.

• Hint: $f_n=\sum_{k=1}^n k^{-1} \mathbb 1_{[k-1, k)}$ where $\mathbb 1_A$ is the indicator function of $A$. – user251257 Aug 13 '16 at 14:17
• @user251257 Can you please explain this example little bit. – Mr. MBB Aug 14 '16 at 3:29

Let $\mathbb 1_A$ denote the indicator function of the subset $A\subseteq \mathbb R$, that is $\mathbb 1_A(x)$ equals $1$ if $x\in A$ and $0$ otherwise.
Further for $n\in\mathbb N$ let $$f_n = \sum_{k=1}^n k^{-1}(\mathbb 1_{[k-1,k)} - \mathbb 1_{[-k, -k+1)})$$ and $$f = \lim_{n\to \infty} f_n.$$
Then, $f_n$ is Riemann integrable. But, for any $a\in \mathbb R$ we have $$\int_a^\infty f = \infty$$ and $$\int_{-\infty}^a f = -\infty.$$ Hence, their sum is not defined and $f$ is not (improper) integrable.