[Math] Prove that a subset of a set of measure zero has measure zero

Question

Thm Prove that a subset of a set of measure zero has measure zero.

I attempted the proof, corrections appreciated.

Pf

Let $A=\{x_1,….,x_N\}$ be a finite set, and let $\epsilon > 0$ be given.

Then $\cup^{\infty}_{i=1}U_N = \{ (x_1 – \frac{\epsilon}{4N} , x_1 + \frac{\epsilon}{4N}),…….,(x_N – \frac{\epsilon}{4N} , x_N + \frac{\epsilon}{4N})\}$

is an open cover of A.

Also, $U_{n_k} = 1,….j$ is a finite subcover of $\cup^{\infty}_{i=1}U_N$

Let $A' = \{x_1,….x_j\}$ be a subset of A.

Since there are N intervals each of measure $\frac{\epsilon}{2N}$ so that our open cover has measure $N \frac{\epsilon}{2N} = \frac{\epsilon}{2} < \epsilon$

Thus are subcover also has measure zero.

Answer 1

You have only shown this for finite sets. In general, use that $A \subset B$ implies $\mu (A) \leq \mu (B)$. This follows from the additivity axiom.