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I'm reading about curves and came across the statement that a curve $v: [0,1]\rightarrow S$ is "differentiable at $L^1$-a.e. point $t \in [0,1]$." $S$ is some metric space on which we're attempting to define differentiable curves.

I understand what it means for a property to hold a.e. (e.g. for $t\in [0,1]$ a.e. with respect to the Lebesgue measure) but not $L^1-a.e.$

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  • $\begingroup$ The velocity of $v$ is defined a.e. and it is in $L^1$? $\endgroup$
    – Mathitis
    Commented Nov 20, 2021 at 10:05
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    $\begingroup$ The notation is unfortunate, but here I don't think $L^1$ refers to the Lebesgue space of integrable functions, rather its referring to one-dimensional Lebesgue measure on the real line. Possibly more standard notations for the Lebesgue measure on the real line might be $\lambda^1$ or $\lambda_1$ or simply $\lambda$, or $m_1$ or just $m$. $\endgroup$
    – peek-a-boo
    Commented Nov 20, 2021 at 10:12
  • $\begingroup$ @peek-a-boo I think you are right!!! $\endgroup$
    – 900edges
    Commented Nov 20, 2021 at 10:15

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