# What does the notation mean?

Given a measure space $(\Omega,\mathcal{F},\upsilon)$ and a $p>0$. What does the following mean? $$\|f\|_p=(\upsilon|f|^p)^{1/p}$$

$$\|f\|_p=\left(\int_\Omega \vert f\vert^p dv\right)^{1/p}$$ One way to justify their shorthand is that integration is like "measuring" the function, so $v\vert f\vert^p$ could be thought of as shorthand for $\int_\Omega\vert f\vert^pdv$.
• Is $|f|^p$ the $p^{th}$ power of the absolute value of the function $f$ ? – N Zhang Apr 8 '13 at 18:04
• so $|f|^p$ is another function, say, $g$, and $\upsilon|f|^p$ is the integral of $g$ with respect to $\upsilon$ ? – N Zhang Apr 8 '13 at 18:10
• Correct, with the integration being over $\Omega$. – icurays1 Apr 8 '13 at 18:12
Indeed, for every measure $v$ and every integrable function $g$, one can use indifferently $$v(g),\qquad\int g\mathrm dv,\qquad\int_\Omega g(\omega)\mathrm dv(\omega),$$ and a few other combinations of the above to denote the integral of the function $g$ with respect to the measure $v$. In particular, $$\|f\|_p=\left(\int_\Omega |f(\omega)|^p\mathrm dv(\omega)\right)^{1/p}=(v(|f|^p))^{1/p}.$$