Convergenza e spazi Lp

[maikkk1]

Ciao a tutti! mi sono bloccato su un esercizio dove probabilmente bisogna applicare Fatou, ma non saprei come. Consigli? Ecco il testo:

mostrare che $AAf in L^p\text{((0,}oo))$ vale $(\lim_{x rarr oo}\text{inf}) xabs(f(x)) = 0$

Vale anche per il $\lim^{_}$? (dimostrazione o controesempio)

Grazie in anticipo!

[javicemarpe]

Probably you wanted to say$\forall f\in L^p$. If I'm right, your question just doesn't make sense. I mean, if $f\in L^p(0,1)$, then it is defined in $(0,1)$. Maybe your question could have some sense if you understand $f$ to be defined as $0$ out of $(0,1)$. But, then, the answer is totally trivial.

Summarising: try to write your question in a better way.

[maikkk1]

Oh sorry! obviously what I wrote didn´t make any sense. I just properly reformulated the question.

[javicemarpe]

Try to do it by contradiction. You just have to write the negation of $\lim \text{inf}_{x\to\infty}x|f(x)|=0$ and to use the fact that your function is in $L^p$ (maybe you'll need some known inequality involving these spaces). I think this hint should be enough.

Try to solve your second question after thinking about the first one.