Let $\{f_n\}_{n=1}^{\infty}$ be a sequence of continuous real-valued functions defined on $\mathbb R$ which converges pointwise to a continuous real-valued function $f$. Which of the following statements are true?

(i) If $0\leq f_n \leq f$ for all $n\in \mathbb N$ then $\displaystyle \lim_{n\to \infty} \int_{-\infty}^{\infty}f_n(t)dt=\int_{-\infty}^{\infty}f(t)dt.$

(ii) If $|f_n(t)|\leq |\sin t|$ for all $t\in \mathbb R$ and for all $n\in \mathbb N,$ then $\displaystyle \lim_{n\to \infty} \int_{-\infty}^{\infty}f_n(t)dt=\int_{-\infty}^{\infty}f(t)dt.$

(i) If $\int_{-\infty}^{\infty} f<\infty$, then we can use DOMINATED CONVERGENCE theorem and can say that the statement is true. But if $\int_{-\infty}^{\infty}f=\infty$ then what can we say about the statement?

(ii) I was not able to do this one.

Note: At the answer-key it's given that (i) is true but (ii) is false.


(1) Suppose $0 \le f_n \le f$ and $\int_{-\infty}^\infty f(t)\; dt = \infty$. Given $N > 0$, there is $M$ such that $\int_{-M}^M f(t)\; dt > N$. By dominated convergence $\int_{-M}^M f_n(t)\; dt \to \int_{-M}^M f(t)\; dt$, so $\int_{-\infty}^\infty f_n(t)\; dt \ge \int_{-M}^M f_n(t)\; dt > N$ for sufficiently large $n$.

(2) Try $f_n(t) = \sin(t)$ for $n \pi < t < (n+1)\pi$, $0$ otherwise.

  • 3
    $\begingroup$ $f$ is stated to be a continuous real-valued function. $\endgroup$ – Robert Israel Jan 10 '19 at 16:25

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