Must any set of positive Lebesgue measure contain a bounded set of positive measure?

Question

Is this statement true?

Any set $A \in \mathbb{R}^k$ with $\lambda^k(A) > 0$ contains a bounded set $K \subseteq A$ with $\lambda(K) > 0$, where $\lambda^k$ denotes the $k$-dimensional Lebesgue measure.

Answer 1

Let $B_n = A \cap [-n,n]^k$ where $A$ is measurable. Then $(B_n)$ is an increasing sequence of bounded measurable sets contained in $A$, and $A = \cup_{n\geq 0} B_n$, hence $\lambda^k(B_n) \to \lambda^k(A)$.