Exam Question in Analysis

Question

this is the third part of an exam question. I needed a little help starting it. Thanks in advance.

Let $f$ be integrable on a measure space $(X, \mu)$ with $\mu(X) = 1$. Show that $\displaystyle ||f||_p \rightarrow \exp \left(\int_X\log |f|d\mu \right)$ as $p \rightarrow 0.$

Hint: Apply a convergence theorem to $|f| - 1 - \dfrac{|f|^p - 1}{p}$.

See: http://math.stackexchange.com/questions/115428/about-a-probability-measure/115460#115460 — Davide Giraudo, Mar 06 '13 at 20:07
Answered in http://math.stackexchange.com/questions/282271/scaled-lp-norm-and-geometric-mean/282311#282311 — abatkai, Mar 06 '13 at 20:33
I understand this proof, but how would you go about proving the statement incorporating the above hint? — sarah, Mar 07 '13 at 03:32

score 1 · Answer 1 · answered Mar 07 '13 at 14:28

1

In order to use the hint, consider $\{p_n\}$ a sequence which converges to $0$ and define $$f_n(x):=|f(x)|-1-\frac{|f|^{p_n}-1}{p_n}.$$ Look at the pointwise convergence of $\{f_n\}$, then try to find a dominating integrable function.

answered Mar 07 '13 at 14:28

Davide Giraudo

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Exam Question in Analysis

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