Positive integral implies positive function?

Question

Let $f:I\to\mathbb{R}$ be a nonnegative, bounded, integrable and continuous function with $I\subseteq\mathbb{R}$ being an interval.

If $\int_I f d\mu=1$, does $\mu(f>0)>0$ hold, where $\mu$ is the Lebesgue measure?

Answer 1

Suppose $\mu \{f>0\}=0.$ Observe

$$\int_I f\,d\mu = \int_{f>0} f\,d\mu + \int_{f=0} f\,d\mu.$$

The first integral on the right is $0$ because integrating over a set of measure $0$ gives $0.$ The second integral is $0$ because the integral of the zero function over any measurable is $0.$ Thus $\int_I f\,d\mu=0,$ contradiction.