always exists. When ${\mathcal{H}}$ contains one hypergraph $H$, then we write $\pi(H)$ instead of $\pi(\{H\})$.

Throughout, we will consider $n$ growing to infinity, and $R$ to be a fixed set (not growing with $n$). Note that $\pi(\mathcal{H})$ agrees with the *usual* definition of $$\displaystyle \pi(\mathcal{H})=\lim_{n\to\infty}\frac{\text{ex}(\mathcal{H},n)}{\binom{n}{k}}$$ when $\mathcal{H}$ is a set of $k$-uniform hypergraphs. The following result is a direct generalization Katona-Nemetz-Simonovit’s theorem [@KNS].

For any family ${\mathcal{H}}$ of hypergraphs with a common edge-type $R$, $\pi(\mathcal{H})$ is well-defined, i.e. the limit $\displaystyle \lim_{n\to\infty}\pi_{n}({\mathcal{H}})$ exists.

It suffices to show that $\pi_n({\mathcal{H}})$, viewed as a sequence in $n$, is decreasing.

Write $R=\{k_1,k_2,...,k_{r}\}$. Let $G_{n}\subseteq K^{R}_{n}$ be a hypergraph with $v(G_{n})=n$ not containing $\mathcal{H}$ and with Lubell value $h_n(G_{n})=\pi_n({\mathcal{H}})$. For any $\ell< n$, consider a random subset $S$ of the vertices of $G$ with size $|S|=\ell$.

Let $G_{n}[S]$ be the induced subgraph of $G_{n}$ (whose vertex set is restricted to $S$). Clearly $$\pi_\ell({\mathcal{H}}) \geq
\mathbb{E}(h_{\ell}(G_{n}[S])).$$ Write $E(G_{n})=E_{k_1}\bigcup
E_{k_2}\bigcup ...\bigcup E_{k_r}$ where $E_{k_i}$ contains all the edges of size $k_{i}$. Note that the expected number of edges of size $k_i$ in $G_{n}[S]$ is precisely $\frac{\binom{\ell}{k_i}}{\binom{n}{k_i}}|E_{k_i}|$. It follows that $$\begin{aligned}
  \pi_\ell({\mathcal{H}}) &\geq \mathbb{E}(h_{\ell}(G_{n}[S])) \\
  &=\sum_{i=1}^{r} \frac{\mathbb{E}(|E_{k_i} \bigcap \binom{S}{k_i}|)}{\binom{\ell}{k_{i}}} \\
  &=\sum_{i=1}^{r} \frac{ \frac{\binom{\ell}{k_{i}}}{\binom{n}{k_i}}|E_{k_i}|}{\binom{\ell}{k_{i}}} \\
  &=\sum_{i=1}^{r} \frac{ |E_{k_i}| 