The Integrals of min(My², M|y|³/(2√n)) Tend to Zero under a Finite Second Moment

In the setting of Measurable Functions and the Lebesgue Integral: Notation, let $M\ge0$ be a real number. For $n\in\mathbb{N}$, read in $\mathbb{R}$ (The Natural Numbers Read in ℝ: a Predicative Condition and a Function from ℕ₀ to ℝ (ii)) with square root $\sqrt{n}$, and for $y\in\mathbb{R}$ let $g_{n}(y)$ be the minimum

$$g_{n}(y)=\min\Bigl(My^{2},\frac{M|y|^{3}}{2\sqrt{n}}\Bigr).$$

Let $\mathcal{L}=\mathrm{Meas}(\mathbb{R},\mathcal{B}(\mathbb{R});\mathbb{R},\mathcal{B}(\mathbb{R}))$ be the set of Borel functions $\mathbb{R}\to\mathbb{R}$.

(i) A sequence of Borel functions. For every $n\in\mathbb{N}$, $y\mapsto g_{n}(y)$ is a function $g_{n}:\mathbb{R}\to\mathbb{R}$ and $g_{n}\in\mathcal{L}$; and $n\mapsto g_{n}$ is a sequence $(g_{n})_{n\in\mathbb{N}}$ in $\mathcal{L}$.

In the remaining clauses, let $\mu$ be a measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ such that the function $y\mapsto y^{2}$ on $\mathbb{R}$ lies in $\mathcal{L}^{1}(\mu)$.

(ii) Integrability. $g_{n}\in\mathcal{L}^{1}(\mu)$ for every $n\in\mathbb{N}$.

(iii) Vanishing integrals. For the integrals of the elements $g_{n}$ of $\mathcal{L}^{1}(\mu)$ ((ii)),

$$\int_{\mathbb{R}}g_{n}\,d\mu\to0 .$$