MHB Show Inclusion of Measures: Hölder's Inequality

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mathmari
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Hey! :o

Let $1 \leq p_1 \leq p_2 \leq +\infty$. Show that in a Lebesgue measurable $E\subset R^d$ with $0<m(E)<+\infty$ we have that $L^{p_2} \subsetneq L^{p_1}$.

Using Hölder's inequality I got that $||f||_{p_1} \leq ||f||_{p_2} \mu (E)^{1/p_1 \cdot q}$.

Is this correct so far?? How could I continue to show that $||f||_{p_1} < ||f||_{p_2}$ ?? (Wondering)

Or is there an other way to show this?? (Wondering)
 
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mathmari said:
How could I continue to show that $||f||_{p_1} < ||f||_{p_2}$ ?? (Wondering)

Or isn't this that we want to show so that $L^{p_2} \subsetneq L^{p_1}$ ?? (Wondering)
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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