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 $d_1$.
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.