Distribution of a conditional expectation I am reading a book on financial mathematics and a Theorem gives a price formula for a Call Option:

$$
\begin{aligned}
\pi_{\text {call }}(t)
&=P(t, S) q(t, S, \mathcal{I})-K P(t, T) q(t, T, \mathcal{I}) \\
\end{aligned}
$$
where $\mathcal{I}=(A(S-T)+\log K, \infty)$, and $q(t, S, d y)$ and $q(t, T, d y)$ denote the $\mathcal{F}_{t}$ conditional distributions of the real-valued random variable $Y=-B(S-T)^{\top} X(T)$ under the $S$ - and $T$-forward measure, respectively.

In the last step of the proof, he comes up with the formula

$$
\pi(t)=P(t, S) \mathbb{Q}^{S}\left[E \mid \mathcal{F}_{t}\right]-K P(t, T) \mathbb{Q}^{T}\left[E \mid \mathcal{F}_{t}\right]
$$
for the exercise event $E=\left\{-B(S-T)^{\top} X(T)>A(S-T)+\log K\right\}$.

I am confused, because the first formula looks like a real value (the measure $q(t,T,\cdot)$ evaluated on the set $\mathcal{I}$) and the second formula looks like a random variable to me ( $\mathbb{Q}^{S}\left[E \mid \mathcal{F}_{t}\right]$ is a random variable).
How do $q(t, T, \mathcal{I})$ and $\mathbb{Q}^{T}\left[E \mid \mathcal{F}_{t}\right]$ coincide? What exactly is a $\mathcal{F}_{t}$ conditional distribution of the real-valued random variable $Y=-B(S-T)^{\top} X(T)$ under $T$-forward measure?
We can rewrite $E$
$$E=\{Y> A(S-T)+\log K\}=\{Y\in (A(S-T)+\log K,\infty)\}=\{Y\in\mathcal{I}\}$$
and the conditional distribution
$$\mathbb{Q}^{T}\left[E \mid \mathcal{F}_{t}\right]=\mathbb{Q}^{T}\left[Y\in\mathcal{I} \mid \mathcal{F}_{t}\right]=\mathbb{Q}^{T}\left[\cdot \mid \mathcal{F}_{t}\right](Y^{-1}(\mathcal{I}))$$
I guess $\mathbb{Q}^{T}\left[\cdot \mid \mathcal{F}_{t}\right]$ can be a probability measure, let's say $Q_1$, in some circumstances? And then we would have the distribution of $Y$ under the measure $Q_1$ ?
 A: I found a footnote on this question. One distinguishes between conditional probability and conditional distribution.

Recall that for every $\mathbb{R}^{n}$-valued random variable $Z$ and sub- $\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$, there exists a regular conditional distribution $\mu(\omega, d z)$ of $Z$ given $\mathcal{G}$. That is, $\mu(\omega, \cdot)$ is a probability measure on $\mathbb{R}^{n}$ for every $\omega \in \Omega, \omega \mapsto \mu(\omega, E)$ is $\mathcal{G}$-measurable for every $E \in \mathcal{B}\left(\mathbb{R}^{n}\right)$, and $\mathbb{E}[f(Z) \mid \mathcal{G}](\omega)=$ $\int_{\mathbb{R}^{n}} f(z) \mu(\omega, d z)$ for all bounded measurable functions $f$, for a.e. $\omega$. See e.g. [Sect. 44, Bauer,H.:Wahrscheinlichkeitstheorie, 5thedn. deGruyterLehrbuch, ISBN 3-11-017236-4]

Especially in the author's notation, we find
$$
\begin{align}
\mathbb{Q}^S[E|\mathcal{F}_t](\cdot)&=\mathbb{Q}^S[Y^{-1}(\mathcal{I})|\mathcal{F}_t](\cdot)\\
&=E_{\mathbb{Q}^S}[1_\mathcal{I}(Y)|\mathcal{F}_t](\cdot)\\
&=\int_\mathbb{R} 1_\mathcal{I}(y)\mu(\cdot,dy)\\
&=\int_\mathbb{R} 1_\mathcal{I}(y)\mathbb{Q}^S[Y^{-1}(dy)|\mathcal{F}_t](\cdot)\\
&=\int_\mathcal{I} \mathbb{Q}^S[Y^{-1}(dy)|\mathcal{F}_t](\cdot)\\
\end{align}$$
where
$$q(t,S,dy)=\mathbb{Q}^S[Y^{-1}(dy)|\mathcal{F}_t]$$
and
$$q(t,S,dy)(\omega)=\mathbb{Q}^S[Y^{-1}(dy)|\mathcal{F}_t](\omega)$$
