There is this problem which I have constructed a plan to prove, and I am stuck. If anyone could see my plan and tell what is wrong about it I would be very thankful.

Let $f: Q \to [0,1]$ be integrable where Q is a rectangle in $\Bbb R^n$. The graph of f is the set $H_f={(x,y)\in R^{n+1}} | x\in Q, y=f(x)) \subset Q\times [0,1] $. Show that $H_f$ is measure zero.

My plan is this: If I can show that a function over $H_f$ is integrable, then then the set $H_f$ is measure zero. $H_f$ is composed of the domain and codomain of the function $f$. $f$ is integrable on $Q$ and maps to $[0,1]$. $Q$ is certainly of measure zero. If $[0,1]$ is measure zero, then $H_f$ would be of measure zero since $H_f$ is the union of the domain and codomain of $f$. So my problem is whether I can assert that $[0,1]$ is measure zero. My idea here is to show that $f$ is invertible, but I do not know how to do that given this conditions.

This is not a homework question

Thanks for your help!


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