Definition of zero test set

Definition of zero test set

设 $A\subset {\mathbb{R}}^2$ is the set of plane points. If any given$e>0$ , there can be at most several closed rectangles

$$I_i,i=1,2,\cdots$$

Make

$$A\subset U_{i⩾1}{I}_i,And\sum _{i⩾1}v(I_i)<ϵ$$

Called $A$ is the zero test set

The basic theorem of zero test set

1. The finite point sets are all zero test sets
2. The subset of the zero test set is the zero test set
3. Several zero test sets can still be a zero test set
4. The sides of the rectangle are the zero test set
5. Let $f$ is$[a,b]$ , thegraph (f) = {(x, f (x)) ∣ x ∈ [a, b]} ⊂ R 2 {\rm graph}(f)=\{( x,f(x))|x\in[a,b]\}\subset\mathbb{R}^2graph(f)={ (x,f(x))∣x∈[a,b]}⊂R2 is the zero test set