What is the difference between the spaces $L^1$($\mu$) and $L^1$(d$\mu$) ? And is one a subset of the other?
$\mu$ is the Lebesgue measure.
2026-04-28 09:48:45.1777369725
What is the difference between the spaces $L^1$($\mu$) and $L^1$(d$\mu$)? And is one a subset of the other?
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There is no difference. Both stand for the Lebesgue space of integrable functions, up to equality $\mu$-almost everywhere.
The inconsistency of notation originates in the notation for the integral of $f$ with respect to $\mu$: it can be written as
$$\int f(x)\,d\mu(x),\quad \int f\,d\mu, \quad \text{or }\quad \int f\,\mu$$ On one hand, mathematicians prefer single-letter names for objects (such as measures), on the other, they are used to $dx$ in integrals. Hence the inconsistency.