I want to show that $\Gamma(R)\times\Gamma(R)$ is a subset of $\Gamma(R^2)$, where $\Gamma(\cdot)$ are the lebesgue sets of $R$ or $R^2$ respectively. What can i do for that and why is it a subset and not equal as one would probably estimate?
To show $\Gamma$(R)^2 is subset of $\Gamma$($R^2$)
$\Gamma(\mathbb{R}) \times \Gamma(\mathbb{R}) \subset \Gamma(\mathbb{R}^2)$, this is by the construction of the Lebesgue measure for $\mathbb{R}^2$.
To show it is a proper subset, just observe the set $\{1\} \times A$ where $A\not\in \Gamma(\mathbb{R})$ has zero Lebesgue measure in $\mathbb{R}^2$, thus $\{1\} \times A \in \Gamma(\mathbb{R}^2)$ and clearly $\{1\} \times A \not\in \Gamma(\mathbb{R}) \times \Gamma(\mathbb{R})$.