Integration Inequality for unbounded vs bounded function

Question

Let $f:\mathbb{R}^d \rightarrow \mathbb{R}$ and $g:\mathbb{R}^d \rightarrow \{0,1\}$. $f$ is unbounded. Is the following true?

$$\int_{x \in \mathbb{R}^d} f(x)\ \mathsf dx \geq \int_{x \in \mathbb{R}^d}g(x)\ \mathsf dx.$$

Answer 1

Take $g=1_{[0,1]^d}$, $f= \sum_{n=0}^\infty n \cdot 1_{\{n\}}$, then $f$ is unbounded, but $\int f = 0$ and $\int g = 1$.