If we think of a boolean function of type $\mathbb{B}^k \to \mathbb{B}$ as a "proposition" about bit strings in $\mathbb{B}^k$, then it amounts to the simplest example of a "distribution" over $\mathbb{B}^k$. Keeping to the field GF(2) for everything in sight, a function of type $(\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B}$ can be thought of as a "higher order proposition", that is, a proposition about propositions about bit strings of length $k$. But it may also be thought of as the simplest example of a distribution over distributions over the space $\mathbb{B}^k$, in other words, a "measure" on distributions over $\mathbb{B}^k$. The link just given exhibits some pictures for the low dimension cases $k = 1, 2$ that may help to support the intuition a bit.
If we think of a boolean function of type $\mathbb{B}^k \to \mathbb{B}$ as a "proposition" about bit strings in $\mathbb{B}^k$, then it amounts to the simplest example of a "distribution" over $\mathbb{B}^k$. Keeping to the field GF(2) for everything in sight, a function of type $(\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B}$ can be thought of as a "higher order proposition", that is, a proposition about propositions about bit strings of length $k$. But it may also be thought of as the simplest example of a distribution over distributions over the space $\mathbb{B}^k$, in other words, a "measure" on distributions over $\mathbb{B}^k$. The link just given exhibits some pictures for the low dimension cases $k = 1, 2$ that may help to support the intuition a bit.
If we think of a boolean function of type $\mathbb{B}^k \to \mathbb{B}$ as a "proposition" about bit strings in $\mathbb{B}^k$, then it amounts to the simplest example of a distribution "distribution" over $\mathbb{B}^k$. Keeping to the field GF(2) for everything in sight, a function of type $(\mathbb{B}^k \to \mathbb{B}) \to \mathbb{B}$ can be thought of as a "higher order proposition", that is, a proposition about propositions about bit strings of length $k$. But it may also be thought of as the simplest example of a distribution over distributions over the space $\mathbb{B}^k$, in other words, a "measure" on distributions over $\mathbb{B}^k$. The link just given exhibits some pictures for the low dimension cases $k = 1, 2$ that may help to support the intuition a bit.