Counterexample for Finding Lebesgue Integrable Function

Question

Prove (or provide a counterexample). There does not exist an integrable function $g$ on $\mathbb{R}^n$ that is

• Nonnegative
• Has measure satisfying $m(${$g > 2$}$)\hspace{.2cm} \geq \frac{1}{2}$
• $\int_{\mathbb{R}^n} g = 1$.

In $\mathbb{R}$, a counterexample would be $g(x) = 3 \cdot \mathbb{1}${$x \in [0, 1/3]$}. However, I'm having difficulties visualizing an extension to $n$-dimensions. Or perhaps there is no counterexample?

Answer 1

Suppose that $g$ has the required properties. Let $A = \{x\in \mathbb{R^n}: g(x)>2\}$ then $m(A)\geq \frac{1}{2}$. Then $g> 2\cdot \mathbb{1}_A$. What can you say about the integral on the right-hand side?

Answer 2

There is no such function. $\int_{g>2} (g-2)dm \leq 1-2m(g>2) \leq 1-2\frac 1 2 =0$. But $\int_{g>2} (g-2)dm >0$.