EXPONENTS OF MODULARED SEMI-ORDERED LINEAR SPACES

Sadayuki Yamamuro
1953 Journal of Faculty of Science Hokkaido University Series I Mathematics  
a set of measurable functions $a(t)(0\leqq t\leqq 1)$ such that $\int_{0}^{1}|a(t)|^{p}dt<+\infty$ , is obviously a moduiared semi-ordered linear space, putting its modular as $m(a)=\int_{0}^{1}|a(t)^{p}|dt$ . $L_{p}$ -space is but an example of the modulared semi-ordered linear space. Moreover we have (3) $\frac{m(\xi a)}{\xi}\leqq\pi(\xi/a)\leqq\frac{m((\xi+\eta)a)}{\eta}$ for any $positi^{\backslash }ve$ number $\xi$ and $r_{/}$ . The right-hand inequality is. a immediate consequence of the
more » ... consequence of the definition, that is, $\pi(\xi/a)\leqq\frac{m((\xi+\eta)a)-m(\xi a)\backslash }{\eta}-\leq\frac{m((\xi+\eta)a)}{\eta}*$ If $m(aa)$ is finite, we have $ORf_{I}ICZ$ -BIRNBAUM[12] and W. H. YOUNG [17]. Remark. To prove the inequality:
doi:10.14492/hokmj/1530864203 fatcat:73j6wialhzehpplg5hbzfh3wse