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I'm a bit confused about how to take the distance between two functions where one function is discontinuous. Supposing we have the $L^1$ metric $d_1$ and $f_n(x) = x^n$ defined over $[0, 1]$. $x^n$ has discontinuous limit function so how would one take the distance under $d_1$?

Would it look like ($x=1$): $$d_1(x^n, 1) = \int_1^1 |x^n - 1|dx$$ and for $0\le x \lt1$ $$d_1(x^n, 0) = \int_0^1 |x^n - 0|dx$$? Thanks, I'm just a little unsure. I want to find out whether $f_n$ converges to its pointwise limit under $d_1$.

Tito
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In general, converges in the $L^p$ norm for $p < \infty$ does not imply pointwise convergence of the sequence - at best, it implies pointwise a.e. convergence of a subsequence.

The limit function here is the zero function: We have $$d_1(x^n, 0) = \int_0^1 x^n dx = \frac{1}{n + 1} \to 0$$ as $n \to \infty$, implying that $x^n \to 0$ in the $d_1$ metric.

It's worth pointing out that the pointwise limit, which is $0$ for $x \in [0,1)$ and $1$ at $x = 1$, is equal to the zero function almost everywhere. So in the Lebesgue space $L^1$, we would identify these two functions as equal.

  • I understand your answer. The problem is I'm studying a general topology course and we're only interested in the metrics coming from $L^1$ - we don't learn about these spaces. So the question asks me to find the pointwise limit and then check whether the sequence converges to its pointwise limit under the $L^1$ metric. That is, I'm not too sure how to approach my question since it first asks to find the pointwise limit? – Tito May 20 '14 at 22:04
  • The pointwise limit of the sequence isn't an element of the underlying space (since it's not continuous), so it doesn't make sense to talk about it in the given metric. –  May 20 '14 at 22:05
  • So then would we say that it doesn't converge to its pointwise limit in $L^1$ or that this question doesn't make sense? – Tito May 20 '14 at 22:07
  • It doesn't have a pointwise limit in the given space ($C[0,1]$; in the larger space $L^1$, it does), but the sequence is convergent in the given metric $d_1$. –  May 20 '14 at 22:08
  • Haha I appreciate your comments - but it doesn't clear the confusion. The question as given to me asks: "Find the pointwise limit $f$ of $f_n$ and investigate if $f_n$ converges to $f$ with respect to the $L^1$-metric." So what's the answer? :P – Tito May 20 '14 at 22:12
  • @McT The pointwise limit is the discontinuous function you found, and it's easy to check that $\int |x^n - f| \to 0$, where $f$ is the discontinuous pointwise limit. My answer was based on the comment that you're restricting your attention to $C[0,1]$. –  May 20 '14 at 22:13
  • I don't doubt your answer :) We are restricting ourselves to C[0,1]. Also could you tell me how it's easy to check that? – Tito May 20 '14 at 22:18
  • You can just compute the integral directly; it's equal to $1 / (n + 1)$. –  May 20 '14 at 22:21
  • :( but the whole reason I'm confused is that the limit function is discontinuous so I don't see how we can pick f = 0 when f isn't 0 everywhere :( – Tito May 20 '14 at 22:23