On the measurability of a family of linear varieties (Q2704119)

The authors study the measurability of planes and lines in the 3-dimensional space \(A_3\). In the affine space \(A_3\) the family of planes and lines of the form NEWLINE\[NEWLINEX=\left(\begin{smallmatrix} x_1\\ x_2\\x_3\end{smallmatrix}\right),\quad B=\left(\begin{smallmatrix} b_1\\ b_2\\b_3\end{smallmatrix}\right)\quad L=\left(\begin{smallmatrix} l_1\\ l_2\\ 1\end{smallmatrix}\right),\quad Q=\left(\begin{smallmatrix} q_1\\ q_2\\ 0\end{smallmatrix}\right),\quad X'=\left(\begin{smallmatrix} x_1\\ x_2\\x_3\end{smallmatrix}\right)NEWLINE\]NEWLINE NEWLINE\[NEWLINEP=(p_{ij}),\quad i,j=1,2,3,\quad\text{det} P\neq 0,\quad ^tA=(\alpha_1,\alpha_2,\alpha_3)NEWLINE\]NEWLINE are considered, where \(b_1,b_2,b_3,l_1,l_2,q_1,q_2\) and \(b_1\alpha_1 + b_2\alpha_2 + b_3\alpha_3\neq 1\). In this case using the matricial form we have NEWLINE\[NEWLINE{\mathcal T}_7:{}^tBX=1,\quad X=Lx_3+Q,NEWLINE\]NEWLINE NEWLINE\[NEWLINEG_{12}:X=PX'+A.NEWLINE\]NEWLINE The problem is to see how to transform the seven parameters of the family \({\mathcal T}_7\) when we apply a transformation \(T\) of the group \(G_{12}\). For other details see the authors' references.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00040].