In Kollár and Mori's Birational Geometry of Algebraic Varieties, the authors say a Cartier divisor is big iff its birational pullback is big (Definition 2.59 below). But I can't understand. Maybe the varieties are proper by definition of big divisor. 

And I know when the varieties are integral normal and proper, this can be done by Zariski main theorem and projectiive formula. Are conditions 'integral' and 'normal' necessary?

Thank you for any answer or comments.