Is every set with sign measure non-negative is a positive set?

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Today, we studied that

a set $A$ is positive set if $\lambda(B)\geq0$ for every measurable subset $B$ of $A$.

a set $A$ is negative set if $\lambda(B)\leq0$ for every measurable subset $B$ of $A$.

a set $A$ is null if $A$ is both positive and negative.

Which made me think is every set $A$ with $\lambda(A)\geq0$ is positive?

Also, is every set $A$ with $\lambda(A)=0$ is null set?