What are the types of rational algebraic fractions?

floymdiT 2021-11-07 Answered
What are the types of rational algebraic fractions?
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Brighton
Answered 2021-11-08 Author has 103 answers
Step 1
There are two types of rational algebraic fractions namely proper algebraic fractions and improper algebraic fractions.
A rational algebraic fraction f(x)g(x) consisting of the numerator with lower degree than the denominator is defined as proper algebraic fractions.
Example: x2x3+1
Step 2
A rational algebraic fraction f(x)g(x) consisting of the numerator with higher degree than the denominator is defined as improper algebraic fractions.
Example: x2+1x
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I have a measure μ that is supported on [ 3 , 3 ] × R . What we are given is that, if we fix the first component i, then μ ( i , ) is a probability measure. Formally (maybe it should integrate to d i or similar, I cannot define this very well):
x R d μ ( i , x ) = 1 i [ 3 , 3 ] .
I find it very hard to understand this tuple-indexing notation. My question is the following: What kind of assumptions do we need to have a result similar to:
( i , x ) [ 3 , 3 ] × R d μ ( i , x ) = i [ 3 , 3 ] d i

I just think that since for any fixed i the measure μ integrates to 1 (on the second dimension -- apologies for my poor terminology), then integrating over all ( i , x ) [ 3 , 3 ] × R should also give simply an 'iterated integral' where we first integrate wrt x R for a fixed i, which will integrate to 1 , and then integrate over i [ 3 , 3 ]. But of course, we cannot define
( i , x ) [ 3 , 3 ] × R d μ ( i , x ) = i [ 3 , 3 ] x R d μ ( i , x ) = i [ 3 , 3 ] d y
where in the red parts I make an abuse of integartion rules.

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