Lebesgue measure of sum $ X + Y$ is infinite

Question

I am given that $X$ is an interval of length $d > 0$ and $y_n$ is a sequence of reals such that $|y_m - y_n| > d$ if $m\neq n$ . If we let $Y =\{y_n: n \in \mathbb{N}\}$ show that $m(X+Y) = \infty$.

Here $m$ denotes the Lebesgue measure.

Answer 1

Observe that: $$X+Y=\bigcup_{n\in\mathbb N}(X+y_n)\tag1$$ where $X+y_n:=\{x+y_n\mid x\in X\}$.

If $z\in (X+y_n)\cap(X+y_m)$ then $a,b\in X$ must exist with $a+y_n=z=b+y_m$ and consequently $|y_n-y_m|=|b-a|\leq d$ and $n=m$.

This tells us that the sets $X+y_n$ and $X+y_m$ are disjoint if $n\neq m$ so that the RHS of $(1)$ is a union of disjoint sets.

Then: $$m(X+Y)=\sum_{n\in\mathbb N}m(X+y_n)=\sum_{n\in\mathbb N}m(X)=+\infty$$

The second equality because the Lebesgue measure is invariant under translations and the second because $m(X)=d>0$.