A Borel Set Whose Lines in One Coordinate Direction Are Null Is Null

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$ and also with the dimension $1$: the Euclidean space $\mathbb{R}^{n}$ with its norm, the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{n})$, Lebesgue measure $\lambda_{n}$, and the notion of a null subset are as fixed there. We identify a point of $\mathbb{R}^{1}$ with its single coordinate, so that $\mathbb{R}^{1}$ and $\mathbb{R}$ are written interchangeably and $\lambda_{1}$ is the Lebesgue measure $\lambda$ on the Borel $\sigma$-algebra of $\mathbb{R}$.

Let $i$ be a natural number with $1\le i\le n$ and let $e_{i}$ be the standard basis vector of $\mathbb{R}^{n}$ whose $i$th coordinate is $1$ and whose other coordinates are $0$.

Let $N\in\mathcal{B}(\mathbb{R}^{n})$ and suppose that for every $y\in\mathbb{R}^{n}$ the set

$$\{\,t\in\mathbb{R}\;:\;y+te_{i}\in N\,\}$$

is $\lambda_{1}$-null. Then the following holds.

1. (The set is null) (i) $\lambda_{n}(N)=0$; in particular $N$ is null.