prove that a Paraboloid is a hausdorff measurable subset

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I'm trying to study for my exam and the question is:

Let

$A=\{ ( x,y,z) R^3 : || (x,y )|| , z=1/2 ||( x,y )||^2\}$

Show that $A$ is a $H^2$ - measurable subset and calculate the surface area.

I don't have a problem computing the surface area but I don't know how to show the first part.

I would be really thankful if someone could give me a hint.