Proof of strong Holder inequality Let $a>1$ and $f,g :\left(0,1\right) \rightarrow \left(0,\infty\right)$ measurable functions, $B$ a measurable subset of $\left(0,1\right)$ such that $$\left(\int_{C} f^2 dt\right)^{1/2} \left(\int_{C} g^2 dt\right)^{1/2} \geq a \int_{C} fg dt$$ for all $C$ measurable subset of $B$. Prove that $B$ has Lebesgue measure zero.  Is the same true if we consider a probability measure and Borel subsets of $\left(0,1\right)$ and Borel functions?

Yes, it's true in more general situations.
Let $\mu$ a positive measure on $X$, and $f,\, g \colon X \to (0,\,\infty)$ measurable. Let $a > 1$. Then every measurable $B$ with $\mu(B) > 0$ contains a measurable $C \subset B$ with
$$\left(\int_C f^2\,d\mu\right)^{1/2} \left(\int_C g^2\,d\mu\right)^{1/2} < a\cdot \int_C fg\,d\mu.$$
For $c > 1$, $n\in \mathbb{Z}$, measurable $M$ with $\mu(M) > 0$, and measurable $h\colon X \to (0,\,\infty)$, let
$$S(c,k,M,h) := \{ x \in M : c^k \leqslant h(x) < c^{k+1}\}.$$
Each $S(c,k,M,h)$ is measurable, and
$$M = \bigcup_{k \in \mathbb{Z}} S(c,k,M,h)$$
where the union is disjoint. Since $\mu(M) > 0$, at least one $S(c,n,M,h)$ has positive measure.
Choose $1 < c < \sqrt{a}$ and set $m_f = c^n$ where $n\in\mathbb{Z}$ is such that $A = S(c,n,B,f)$ has positive measure. Let $k \in\mathbb{Z}$ such that $C = S(c,k,A,g)$ has positive measure, and set $m_g = c^k$.
On $C$, we have $m_f \leqslant f(x) < c\cdot m_f$ and $m_g \leqslant g(x) < c\cdot m_g$, hence
$$\begin{align}
\int_C fg\,d\mu &\geqslant \int_C m_f\cdot m_g\, d\mu = m_f m_g \cdot \mu(C),\\
\left(\int_C f^2\,d\mu\right)^{1/2} \left(\int_C g^2\,d\mu\right)^{1/2}
&< \left(\int_C(c\cdot m_f)^2\, d\mu\right)^{1/2} \left(\int_C (c\cdot m_g)^2\, d\mu\right)^{1/2}\\
&= c\cdot m_f \sqrt{\mu(C)} \cdot c\cdot m_g \sqrt{\mu(C)}\\
&= c^2\cdot m_f m_g\cdot \mu(C)\\
&< a\cdot m_f m_g\cdot \mu(C)\\
&\leqslant a \int_C fg\,d\mu.
\end{align}$$
