If $f, g : (\Bbb C^n, 0) \to (\Bbb C, 0)$ are two germs of smooth functions then if you can change coordinates to write $f(x_1, \cdots, x_n) + x_{n+1}^2 + \cdots + x_k^2$, germ of a smooth function at $0$ in $\Bbb C^{n+k}$, as $g(y_1, \cdots, y_n) + y_{n+1}^2 + \cdots + y_k^2$, then you can change coordinates to write $f$ as $g$ as well.
When I say change coordinates to write A as B I mean the germs A and B are in the same $\text{DIFF}(\Bbb C^n, 0)$-orbit, where this is the group of germs of diffeomorphisms $(\Bbb C^n, 0) \to (\Bbb C^n, 0)$, acting on all germs in the obvious way
I'll look at your proof in greater detail later, @Leaky. Thanks again
@JoeShmo: Vakil is still extremely commutative algebra heavy. Try Shafarevich for a little lighter, or, for plenty of examples and classical approach, Joe Harris's introductory book (not Griffiths/Harris, although I love that).
In a PDF, the probability is the area under the curve. This means that the probability of any particular value occurring is either zero or a vanishingly small limit. However the probability of some event occurring is 1. How can these two things be compatible? It looks almost as if the first is universally quantified over while the second is existentially quantifying over the negation of what the first is.
It's easier to consider for probabilities case, because we can say things like "the probability that some event will occur is 1", which we can't do the rephrasing.
I understand the rephrasing in the sense of summing lots of infinitesimal quantities, but I can't seem to extend this definition to probability.
the fallacy is in your interpretation. probability just gives you proportions of areas (out of $1$ unit of area). You chose to interpret it so that it means that a certain event should occur -- mind you, I'm not saying that that's unreasonable, but the buy-in is that then it follows that the probability of any one (discrete) event occurring is $0$
I would say that it is unreasonable, as you could express that some event occurs as $\exists x P(x)=1$, and that any particular even occurs as $\forall x P(x)=lim_{k\rightarrow 0} k$.
@Thorgott I am unfamiliar with probability measures as well, but the Wikipedia page on them is surprisingly brief and clear. How would I represent the probability of a particular event occurring in the same notation?
Still, this new explanation just seems to be a rewording of the original question. The probability of some event occurring from the event space is still 1, and the probability of a particular event occurring still isn't 1.
Unless $\mathbb{P}(A)=1$ for the some particular $A$.
if you throw a coin, you will always get heads or tails (let's assume we live in an ideal world where coins don't fall on the side), but it's not like you will always get head or always get tail
But aren't the two statements negations of another? $\exists x\in \Omega P(x)=1$, and $\forall x\in \Omega P(x)\neq 1$. How would I express that the probability of some event occurring from the event space is 1 in symbolic logic? I think this is the heart of my problem.
@Thorgott I can see that my first statement is still falling for the same error mentioned before, but I can't think of another way to write it in symbolic logic.
@Thorgott It is a goal of mine to be able to write it in symbolic logic, because then I can precisely translate it to English.
So my vague statement "The probability of some event occurring from the event space is still 1" could be rephrased into something precise and correct.
Then what would $\mathbb{P}(A)\neq 1$, where $A$ is a particular event, translate to? That the probability of a particular event occurring is not 1? If so, doesn't that seem like a problem?
Probability is about how probable a certain event is, not whether it occurs or not in a specific iteration
The probability of flipping heads in a coin toss is 1/2, because it is just as probable to flip heads as it is to flip tails. This is universal. It does not care about whether you flip heads or tails in one specific coin toss.
@Thorgott Even the second part of this sentence isn't making sense to me. "not whether it occurs or not in a specific iteration" seems to be claiming that it is not the case, that an event will occur, or it will not occur. But this is a contradiction.
No, when you carry out an experiment, such as a coin toss, an event either does occur or does not occur. But we're not concerned with the deterministic post hoc side of things, but the probabilistic pre hoc side of things.
We don't even flip the coin, we just wonder what would happen if we flipped it and how likely each of the possible outcomes is.
My problem is, the event has to either occur, or not occur. If it occurs, then the probability that it occurs is 1. If it does not occur, then the probability that it occurs is 0. But this completely excludes every other probability.
@Thorgott Thank you for your patience explaining everything to me. I haven't reached an understanding yet, but I sincerely appreciate all the help you have given me. This goes for @JoeShmo and @AlessandroCodenotti too. I plan to ask a question on the main site to help clarify my understanding regarding this.
