Measurement Theory – With what formalism can random variables be treated algebraically like real or complex numbers?

We all know that there is meaning when we have a variable x – for example –
$$ y = e ^ x $$,

And we all know how to manipulate and calculate this algebraically. For example when

$$ y_1 = e ^ {x_1} $$


$$ y_2 = e ^ {x_2} $$


$$ y_1y_2 = e ^ {x_1 + x_2} $$

Random variables, however, are a completely different beast. Ultimately, we all know intuitively what we mean when we draw a PDF, or more precisely the CDF, which itself is an algebraic function map $[-infty, infty]$ to $[0,1]$ as part of it defines a random variable.

But what is the actual math that makes this definition of a camper actually algebraic, so that we can happily write things like above

$$ Y_1 = e ^ {X_1} $$


$$ Y_2 = e ^ {X_2} $$


$$ Y_1Y_2 = e ^ {X_1 + X_2} $$

from where $ X_1 $ and $ X_2 $ are random variables – and come through it ???

real analysis – $ f $ is Lipschitz and $ X $ is zero. Show that $ f (X) $ is also zero.

If $ f[a,b] to mathbb {R} $ is Lipchitz and $ X subset [a,b]$ has zero measure show that $ f (X) $ also has the measure zero.

What I did:

As $ X $ has measured zero, $ forall epsilon> 0 $ There is a countable union of open intervals $ I_i $ s.t. $$ X subset cup_ {i = 1} ^ infty I_i $$ and $ sum_ {i = 1} ^ infty | I_i | < epsilon. $ As $ f $ is continuous $ f (I_i) = J_i $ is an interval and $ f (X) subset cup_ {i = 1} ^ infty J_i $, As $ f $ is Lipschitz $ | J_i | leq k | I_i | $, for some $ k <0. $ Now $ sum_ {i = 1} ^ infty | J_i | <k sum_ {i = 1} ^ infty | I_i | <k epsilon, $ and so you can take $ epsilon = frac { epsilon_1} {k} $ break up.


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Real householder Elie Hirschfeld's sons sue him for $ 70 million – everything else

The children of a billionaire house magnate sue their father and claim he refuses to disprove their legitimate disgrace about the employee's assets.

The outcast children of Elie Hirschfeld, a family member, sue her father for $ 50 million ($ 70 million).

His three sons, Benjamin (21), Jonathan (20) and Matthew (19), said their fathers had transferred their interests due to direct or indirect home ownership changes after they became sexually mature in 2009, following their divorce from their mother Susan ,

But now he is doing everything in his facility to save her from the money, the papers in the charge of the Supreme Court of Manhattan of her father, the son of the beleaguered Abe Hirschfeld, the colorful car park magnate, to whom the Post belonged to chaos 16 Days in 1993.

Taking into account the fact that Elie is not talking about his three children, who have all reached the age of majority lately, Elie now refuses to take on the Raptors' Rapture in these properties or to take into account the pensions that these properties have earned. after these interests have been irrevocably transferred to the plaintiffs, the court papers are retrieved.

Elie has hidden these interests Elie Hirschfeld children and continues to disguise these interests in order to deceive the plaintiffs. In fact, a note from Elie contains that time [o]The only responsibility is that children sue me higher.

69-year-old Elie has allegedly refused to designate the trio, taking into account recommendations on interests, income, and management of the various residential and commercial properties in Manhattan and New Jersey.

The plaintiffs now have their votes and are confident that they are holding together from their father, which they rightly understand, according to the court document.

The trio hurts various percentages of the particular home, leading to trust, in bringing together criminal offenses to pay damages of $ 50 million.

In a post at The Post, Elie said the allegations were at best unfounded.

The purest show for these unfounded claims, which dates back to the peak of 10 years, is that my sons and their lawyer are clearly misinformed, said Elie. Unfortunately, the probable excuse is that the allegations in a series of concrete struggles involving the controversial divorce between me and my son's mother are erratic.



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co.combinatorics – Taylor's polynomials and loss of real roots

Real rooting, logarithmic concavity and unimodality are intertwined properties. In that light, I was asked to ask the following question.

Suppose the roots of a polynomial $ p (x) $ are all real and $ p (0)> 0 $, Fix a whole number $ k geq0 $ and look at the function $ f = frac1 {p ^ {2k + 1}} $, I like to look at the partial sums (polynomial)
$$ sum_ {j = 0} ^ {2k} frac {f ^ {(j)} (0)} {j!} x ^ j; tag1 $$
from where $ f ^ {(j)} $ Does that mean $ j $-th derivative.

QUESTION. Is it true that the polynomial in (1) has no real roots?