pointwise convergence

**derek walcott** · March 28th 2010, 09:42 PM

I need help with this problem.

Give an example of a sequence of integrable functions fn:[a,b]-->R convergent (pointwise) to a non-integrable function f:[a,b]-->R.

**Tinyboss** · March 29th 2010, 05:24 AM

What kind of integration? Riemann, Lebesgue, something else?

For Riemann integration, enumerate the rationals in the interval as $\{q_i\}_{i=1}^\infty$ , then let

$f_n(x)=\begin{cases}1&x\in\{q_1,\dots,q_n\}\\0&\te xt{otherwise.}\end{cases}$

Then $\int f_n=0$ for every finite n, but the limit function is not integrable.

**Laurent** · March 29th 2010, 07:06 AM

Originally Posted by derek walcott

I need help with this problem.

Give an example of a sequence of integrable functions fn:[a,b]-->R convergent (pointwise) to a non-integrable function f:[a,b]-->R.

For any integral: choose a non-integrable nonnegative function $f:[a,b]\to\mathbb{R}_+$ (any one you want), then $f_n=\min(f,n)$ is bounded by $n$ , hence integrable on $[a,b]$ , and converges pointwise to $f$ .

**Failure** · April 2nd 2010, 11:03 AM

Originally Posted by Laurent

For any integral: choose a non-integrable nonnegative function $f:[a,b]\to\mathbb{R}_+$ (any one you want), then $f_n=\min(f,n)$ is bounded by $n$ , hence integrable on $[a,b]$ , and converges pointwise to $f$ .

I don't think that from the assumption that f be non-negative, non-integrable it follows that $f_n =\min(f,n)$ is integrable.
For the Riemann integral it is easy to give an example of a non-negative and bounded function that is non-integrable (e.g. the characteristic function of $\mathbb{Q}$ ), and in the case of the Lebesgue integral we must consider the possibility that f might not even be measurable.

So, I think, one would have to be a bit more specific about the cause of the non-integrability of f.

**Laurent** · April 3rd 2010, 11:07 AM

Originally Posted by Failure

I don't think that from the assumption that f be non-negative, non-integrable it follows that $f_n =\min(f,n)$ is integrable.
For the Riemann integral it is easy to give an example of a non-negative and bounded function that is non-integrable (e.g. the characteristic function of $\mathbb{Q}$ ), and in the case of the Lebesgue integral we must consider the possibility that f might not even be measurable.

So, I think, one would have to be a bit more specific about the cause of the non-integrability of f.

Right, I meant: "choose a measurable non-integrable function $f$ ". Since $f$ is the limit of $f_n$ , it has to be measurable.

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