Every language that is reducible to a language in $\Sigma_i^p$ is also in $\Sigma_i^p$ . How?
First of all, there are many reduction concepts, but I assume you mean the polynomial reduction ($\le_p$).

The complexity class $\Sigma_{k}^{p}$ is recursively defined as follows:
\begin{align}
\Sigma_{0}^{p} &amp; := P \\
\Sigma_{k+1}^{p} &amp; := P^{\Sigma_{k}^{p}}
\end{align}

Let $A$ and $B$ be two languages over alphabets $\Sigma_A$ and $\Sigma_B$, respectively. Let $k \in \mathbb{N}_0$, $B \in \Sigma_{k}^{p}$ and $A \le_p B$. Since $A$ is polynomially reducible to $B$, there exists a function $f: \Sigma_A^\star \rightarrow \Sigma_B^\star$ (computable in polynomial time), such that $x \in A \Leftrightarrow f(x) \in B$. So we have an algorithm that decides $A$, which first transforms the input for $A$ to an input for $B$ using $f$, and then uses the algorithm for $B$ on the transformed instance. Therefore, $A$ is in $max\left\lbrace P, \Sigma_k^p \right\rbrace$.

If $k = 0$, $\Sigma_k^p = \Sigma_0^p = P$, then $A$ is also in $P = \Sigma_0^p$, since the transformation requires at most polynomial time and $B$ is in $P$.

If $k &gt; 0$, the transformation is negligible and $A \in \Sigma_k^p$.