Is it true that for an infinite index set $I$, we have that $L_{1}([0,1]^{I}, \mathbb{R})$ can be written as the infinite direct sum of $L_{1}([0,1], \mathbb{R})$, i.e. $$L_{1}([0,1]^{I}, \mathbb{R})=\bigoplus_{l_{1}, I}L_{1}([0,1], \mathbb{R})? $$ Edited in from comment: Sorry if this is confusing, in this case I mean $[0, 1]$ as the compact unit interval, $I$ is an infinite set and $\bigoplus_{l_1, I}$ is the infinite $l_1$ sum over $I$.

# Isomorphy between Lebesgue space $L_1$ and the $l_1$ sum of $L_1[0,1]$ spaces

user44155

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