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Beta
 Post subject: Limit
PostPosted: Tue, 19 Jul 2011 03:14:53 UTC 
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Show that
\lim_{q\to 0^+} \frac{1}{q}\left(\int_E |f|^q -1\right)=\int_E \log|f|,
where m(E)=1.
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outermeasure
 Post subject: Re: Limit
PostPosted: Tue, 19 Jul 2011 07:54:08 UTC 
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Beta wrote:
Show that
\lim_{q\to 0^+} \frac{1}{q}\left(\int_E |f|^q -1\right)=\int_E \log|f|,
where m(E)=1.


m(E)=1 is irrelevant. Apply Lebesgue's monotone convergence theorem.

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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Beta
 Post subject:
PostPosted: Wed, 20 Jul 2011 10:17:02 UTC 
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How the log comes into place?
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outermeasure
 Post subject:
PostPosted: Wed, 20 Jul 2011 11:22:40 UTC 
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Beta wrote:
How the log comes into place?


From \displaystyle\log(x)=\lim_{q\downarrow 0}\frac{x^q-1}{q} (x\geq 0).

_________________
\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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Beta
 Post subject:
PostPosted: Wed, 20 Jul 2011 13:49:27 UTC 
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OK got it. Thanks!
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