This paper is a translation of the the following paper into Persian: [Alfonso Villani, Measurable functions with a given set of integrability exponents, Amer. Math. Monthly, 118 (2011) 77–82.]
Given a measure space $(\Omega,\mathcal{A},\mu)$, it is well known that for every $\mathcal{A}$-measurable function $f:\Omega\rightarrow\mathbb{R}$ the set $\mathcal{E}(f)=\{p\in(0,+\infty)\,:\,f\in\mathcal{L}^p(\mu)\}$ is always an interval, possibly degenerate, but, in general, it cannot be any given interval $I \subseteq (0,+\infty)$. Thus we consider the problem of characterizing those measure spaces for which $\mathcal{E}(f)$ can be an arbitrary subinterval of $(0,+\infty)$. We show that they are precisely the measure spaces such that there is no inclusion between different $\mathcal{L}^p(\mu)$ spaces.
Dehghani, M. and Kazemi, R. (2017). Measurable functions with a given set of integrability exponents. Mathematics and Society, 2(1), 45-50. doi: 10.22108/msci.2017.13183
MLA
Dehghani, M. , and Kazemi, R. . "Measurable functions with a given set of integrability exponents", Mathematics and Society, 2, 1, 2017, 45-50. doi: 10.22108/msci.2017.13183
HARVARD
Dehghani, M., Kazemi, R. (2017). 'Measurable functions with a given set of integrability exponents', Mathematics and Society, 2(1), pp. 45-50. doi: 10.22108/msci.2017.13183
CHICAGO
M. Dehghani and R. Kazemi, "Measurable functions with a given set of integrability exponents," Mathematics and Society, 2 1 (2017): 45-50, doi: 10.22108/msci.2017.13183
VANCOUVER
Dehghani, M., Kazemi, R. Measurable functions with a given set of integrability exponents. Mathematics and Society, 2017; 2(1): 45-50. doi: 10.22108/msci.2017.13183
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