A Measurable Real Function Has Null Level Sets at Levels Arbitrarily Close to Any Point, for a Finite Measure

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation and Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity, let $(X,\mathcal{F},\mu)$ be a measure space with $\mu$ finite, and let $g:X\to\mathbb{R}$ be measurable. For $t\in\mathbb{R}$, $\{g=t\}=\{x\in X:g(x)=t\}$ belongs to $\mathcal{F}$ (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (v)).

(i) Null levels in every interval. For all $a,b\in\mathbb{R}$ with $a<b$ there is $t\in\mathbb{R}$ with $a<t<b$ and

$$\mu(\{g=t\})=0 .$$

(ii) Null levels converging from above. For every $a\in\mathbb{R}$ there is a sequence $(t_{k})$ in $\mathbb{R}$ with $a<t_{k}<a+1/k$ and $\mu(\{g=t_{k}\})=0$ for every $k\in\mathbb{N}$.