I am stuyding for the exam and I need a little help with the following:
Let $m^k$ denote the kth-Lebesgue measure and $H^k$ the kth-Hausdorff measure.
Let $f(x, y, z) = x^2y^3z^4$.
Calculate:
$$\int_Afdm^3$$ where $A=\{(x,y,z) \in \mathbb{R^3} : 0 < x < y < z < 1\}$
and $$\int_BfdH^3$$where $B=\{(x,y,z) \in \mathbb{R^3} : 0 < x = y < z < 1\}$
Here is my solution for the first one:
$A=\{(x,y,z) \in \mathbb{R^3} : 0 < x < y < z < 1\} = \{x \in\mathbb{R} : 0 < x < y\}\times\{y \in\mathbb{R} : 0 < y < z\}\times\{z \in\mathbb{R} : 0 < z < 1\}$
Now I apply Fubini and get: $$\int_Afdm^3 = \int_{(0, 1)}\int_{(0, z)}\int_{(0, y)}x^2y^3z^4m(x)m(y)m(z) = \frac{1}{252}$$
Is this correct and are the steps I made valid? And how do I calculate the second one? Is there an easier way than working with the definition?
($H^k(E) = c \lim_{\delta\to 0}\inf\{\sum_{h}diam(F_h) : E \subset \cup_hF_h , diam(F_h) < \delta\}$, with $c$ a constant)