TITLE: Condition on sigma algebra
QUESTION [6 upvotes]: Say I toss two distinct coins. 
Let A be event: there are two heads, $\{HH\}$
Let $\sigma$ be a sigma algebra $\big\{\emptyset,\Omega,\{HH,HT\},\{TT,TH\}\big\}$
How does one understand $P(A|\sigma)$, $E(A|\sigma)$? 
How does the meaning differ if sigma algebra is chosen differently, say trivial, or full algebra?

REPLY [5 votes]: The following are the definitions of conditional expectation and conditional probability, could you fit your case in?

Definition 1: If $\mathcal F \subseteq \mathcal G$ are two $\sigma$-fields, and $X$ a $\mathcal G$-measurable integrable random variable, then $\mathbb E[X | \mathcal F]$ is defined as any $\mathcal F$-measurable random variable $Y$, such that $\mathbb E[Y;A]=\mathbb E[X;A]$ for every $A \in \mathcal F$. Here $\mathbb E[X;A]$ is a notation for $\int_AX\,d\mathbb P$.
Definition 2: We define conditional probability as $\mathbb P(A | \mathcal F)=
\mathbb E[1_A|\mathcal F]$.

From above definition, such r.v. of $Y$ is guaranteed to exist, and is unique up to a.s. equivalence - this is guaranteed by one version of Radon-Nikodym Theorem (i.e. for finite positive $\mathbb P$, and finite signed $\mathbb Q(A) = \mathbb E[X;A] \ll \mathbb P$, the conditional expectation as a Radon-Nikodym Derivative exits and is unique up to a.s.)