Calvo Aggregate Price Dynamics This question actually comes from economics but, AS USUAL, only the name of the variable and idea (price) is from economics there. The technique used is pure math and since economists are mostly sloppy when explaining the math in the background I decided to ask it here to mathematicians. So it goes as follows: 
There are a continuum number of firms over the interval $ [0,1] $ indexed by $ i $. Time framework is discrete. In each time period $t$, only $ (1-\theta) $ fraction of firms get the chance of setting (optimizing) their prices (this means nothing else but setting price $P_t(i) = P_t(i)^\star $ that maximizes its profit). The rest $ \theta $ fraction leave there prices $ P_t(i) = P_{t-1}(i) $. Since each firm $ i $ is identical each firm who gets the chance to set the price will set identical price $ P_t(i) = P_t^\star $.
Aggregate Price Level in each period  $t$ is defined in the following way:
$$ \mathbb{P_t} = \bigg(\int_{0}^{1} {P_t(i)}^{1-\epsilon} di \bigg) ^\frac{1}{1-\epsilon} $$ 
Given this framework the book Monetary Policy Inflation and Business Cycle by Gali(2008) chapter 3 page 62 explains  Aggregate Price Level Dynamics in the following passage: 
Let $ S(t) \subset [0,1] $  represent the set of firms not reoptimizing their posted price in period $t$. Using the definition of the aggregate price level and the fact that all firms resetting the prices will choose an identical price $ P_t^* $,
$$   \mathbb{P_t} = \bigg[\int_{S(t)} P_{t-1}(i)^{1-\epsilon} di + (1-\theta)(P_t^*)^{1-\epsilon} \bigg] ^\frac{1}{1-\epsilon} = \bigg[ \theta (\mathbb{P_{t-1})^{1-\epsilon}} + (1- \theta ) (P_t^*)^{1-\epsilon} \bigg]^{\frac{1}{1-\epsilon}}$$ 
where the second equality follows from the fact that the distribution of prices among firms not adjusting in period $t$ corresponds to the distribution of effective prices in period $t-1$, though with total mass reduced to $ \theta $.
I have the problem in understanding this second equality where it substitutes integral with $\theta (\mathbb{P_{t-1})^{1-\epsilon}}$. Can anybody explain it? Does the above explanation that the author provides make sense to you? 
 A: After several hours of research I finally found out (thanks to commented book by  Satish Ramanathan  in the comment section of the question) that the author skips several steps to derive the Aggregate Price Level in period $ t $. For those who love deep understanding and will come to the same question in the future or those who will blindly use that equality here, I will provide those steps and detailed derivation of how to get to the following equality: 
$$ \mathbb{P_t} = \bigg(\int_{0}^{1} {P_t(i)}^{1-\epsilon} di \bigg) ^\frac{1}{1-\epsilon} = \bigg[ \theta (\mathbb{P_{t-1})^{1-\epsilon}} + (1- \theta ) (P_t^*)^{1-\epsilon} \bigg]^{\frac{1}{1-\epsilon}} $$ 
Before that let's remember that:
For those firms who set their price in period $t$, $P_t(i) = P_t^*$ and the length of the interval over which $P_t(i) = P_t^*$ will be $ 1-\theta $. 
For those firms who set their price in period $t-1$ and then maintained the same price in period $ t $, $P_t(i) = P_{t-1}^*$ and the length of the interval over which $ P_t(i) = P_{t-1}^* $ will be $\theta(1-\theta) $.
For those firms who set their price in period $t-2$ and then maintained the same price till and in period $ t $,$ P_t(i) = P_{t-2}^* $  and the length of the interval over which  $ P_t(i) = P_{t-2}^* $ will be $\theta^2(1-\theta) $
For those firms who set their price in period $t-3$ and then maintained the same price till and in period $ t $,$ P_t(i) = P_{t-3}^* $  and the length of the interval over which  $ P_t(i) = P_{t-3}^* $ will be $\theta^3(1-\theta) $
We can generalize this and for those firms who set their price in period $t-j$ and then maintained the same price till and in period $ t $,$ P_t(i) = P_{t-j}^* $  and the length of the interval over which  $ P_t(i) = P_{t-j}^* $ will be $\theta^j(1-\theta) $
Since $ \sum_{j=o}^{\infty} {\theta^j (1-\theta)} = 1$,  we can then write 
$$ \mathbb{P_t} = \bigg(\int_{0}^{1} {P_t(i)}^{1-\epsilon} di \bigg) ^\frac{1}{1-\epsilon} =\bigg ( \sum_{j=0}^{\infty} {(1-\theta)\theta^j}{P_{t-j}^*}^{1-\epsilon} \bigg)^\frac{1}{1-\epsilon} $$
Using the same transformation  
$$ \mathbb{P_{t-1}} = \bigg(\int_{0}^{1} {P_{t-1}(i)}^{1-\epsilon} di \bigg) ^\frac{1}{1-\epsilon} = \bigg( \sum_{j=0}^{\infty} {(1-\theta)\theta^j}{P_{t-1-j}^*}^{1-\epsilon} \bigg)^\frac{1}{1-\epsilon} $$ 
this implies that 
$$ \mathbb{P_{t-1}}^{1-\epsilon} = \bigg(\int_{0}^{1} {P_{t-1}(i)}^{1-\epsilon} di \bigg) ^\frac{1}{1-\epsilon} = \bigg( \sum_{j=0}^{\infty} {(1-\theta)\theta^j}{P_{t-1-j}^*}^{1-\epsilon} \bigg) $$ 
now multiply both sides by $ \theta$ 
$$ \theta \mathbb{P_{t-1}}^{1-\epsilon} = \bigg(\int_{0}^{1} {P_{t-1}(i)}^{1-\epsilon} di \bigg) ^\frac{1}{1-\epsilon} = \bigg( \sum_{j=0}^{\infty} {(1-\theta)\theta^{j+1}}{P_{t-1-j}^*}^{1-\epsilon} \bigg) $$
Notice that we can rewrite $\mathbb{P_t} $ using $ \mathbb{P_{t-1}} $:
$$ \mathbb{P_t} = \bigg ( \sum_{j=0}^{\infty} {(1-\theta)\theta^j}{P_{t-j}^*}^{1-\epsilon} \bigg)^\frac{1}{1-\epsilon} = \bigg ( (1-\theta) P_t^* + \sum_{j=0}^{\infty} {(1-\theta)\theta^{j+1}}{P_{t-j-1}^*}^{1-\epsilon} \bigg)^\frac{1}{1-\epsilon} =\bigg( (1-\theta) {P_t^*}^{1-\epsilon} + \theta {\mathbb{P_{t-1}}}^{1-\epsilon} \bigg)^{\frac{1}{1-\epsilon}} $$
For those expert in math please edit if you see the lack of rigor. 
