what is the difference between $P(H)+P(T)=1$ and $P(\{H\})+P(\{T\})=1$? both following 2 equations are to indicate that the probability of either heads or tails, is 1. 
$P(H)+P(T)=1$
$P(\{H\})+P(\{T\})=1$
the second one comes from wiki Probability_axioms 
what is the difference between these 2 notations?
is the measure theoretical perspective the only difference?
 A: $\{H\}$ and $\{T\}$ are subsets of the two-element event space $\{H,T\}$ and so $P(\{H\})$ refers to the probability that the outcome is $\{H\}\subseteq\{H,T\}$ and $P(\{T\})$ refers to the probability that the outcome is $\{T\}\subseteq \{H,T\}.$ From a set-theoretic point of view, you could also note that since $P(\{H\})$ and $P(\{T\})$ are independent, $P(\{H\})+P(\{T\}) = P(\{H\}\cup\{T\}) = P(\{H,T\})=1.$
P({H}) is simply a more formal way of writing P(H), as pointed out in the comments to your question
A: the notation is ambiguous. is a functional notation but incorporates the result in the argument of the function.
P(H) represents the probability to obtain heads after one coin toss.
P({H}) represents the function that models the probability to obtain heads after any coin toss in a series of coin toss.
P(H)+P(T)=1 means that the chance of getting heads or tail is 1 during the same event of a coin toss.
P({H})+P({T})=1 means that the chance of getting heads on current toss or tail on any other toss past, current or future is still 1. This sentence is false.
