$\int_0^1f(x)dx = 2, \int_0^1g(x)dx = 1, \text{and} \int_0^1[f(x)]^2 dx ≤ C$ for some constant $C > 4.$

Question

Suppose $f$ and $g$ are nonnegative measurable functions on the interval $[0,1],$ with the properties

$$\int_0^1 f(x)\,dx = 2, \int_0^1g(x)\,dx = 1, \text{ and }\int_0^1[f(x)]^2 dx \le C$$

for some constant $C > 4.$ Let $E = \{x∈[0,1]:f(x)>g(x)\}$.

Show that $E$ has measure $m(E) \ge 1/C.$

I am not sure what I should use for this. Maybe Holder? Any suggestions?

Answer 1

Suppose $m(E) < \frac 1 C$. Then we have

\begin{align*} \int_0^1 f &= \int_E f + \int_{E^c} f \\ &\le \left(\int_E f^2 \int_E 1\right)^{1/2} + \int_{E^c} g \\ &\le \left(|E| \int_0^1 f^2\right)^{1/2} + \int_{E^c} g \\ &< \Big(\frac 1 C \cdot C)^{1/2} + \int_0^1 g \\ &= 2 \end{align*}

a contradiction. Notice how Holder's inequality, together with the fact that $g \ge f$ on $E^c$, are used in line 2.