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Write $L^1$ (respectively $L^\infty$) for the set of integrable (respectively essentially bounded) functions on $[0, 1]$, with respect to the usual Lebesgue measure $\mu$. Given a non null measurable subset $A$ of $[0, 1]$, define the measure space $\mathbf{A} := (A, \mathcal F_A, \nu_A)$ where $\mathcal F_A $ is the induced sigma algebra, and $\nu_A$ is the normalised induced measure given by $\nu_A (E) = \frac{\mu(E \cap A)}{\mu(A)}$. Given any $f \in L^1$ and a set $A$ as above, define $f_A$ to be the function on $\mathbf{A}$ given by $f_A (x) = \frac{ f(x)}{\mu(A)}$. Let $\mathfrak H$ be a family of non null measurable subsets of $[0, 1]$. Find necessary and sufficient conditions on $\mathfrak H$ such that for every $f$ in $L^1$ the following statement holds: $f$ is in $L^ \infty$ if and only if the family $\{f_A\}_{A \in \mathfrak H}$ is uniformly integrable.
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Sun, 04 Apr 2021 01:13 GMT