nyuuzyou/wikis huggingface dataset:
Vertices of Equilateral Triangle in Complex Plane/Sufficient Condition

== Theorem ==
Let $z_1$, $z_2$ and $z_3$ be complex numbers.
Let $z_1$, $z_2$ and $z_3$ represent on the complex plane the vertices of an equilateral triangle.
Then:
:${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$
=== Corollary ===
== Proof ==
:
Let $T$ be the equilateral triangle whose vertices are $z_1$, $z_2$ and $z_3$.
We have that $z_2 - z_1$ and $z_3 - z_1$ are two sides of $T$ which meet at $z_1$.
From the geometry of $T$ it follows that $z_2 - z_1$ is at an angle of $\pi/3$ to $z_3 - z_1$.
Similarly, $z_1 - z_3$ and $z_2 - z_3$ are two sides of $T$ which meet at $z_3$.
From the geometry of $T$ it follows that $z_1 - z_3$ is at an angle of $\pi / 3$ to $z_2 - z_3$.
From Complex Multiplication as Geometrical Transformation/Corollary:
:$(1): \quad z_2 - z_1 = e^{i \pi / 3} \left({z_3 - z_1}\right)$
:$(2): \quad z_1 - z_3 = e^{i \pi / 3} \left({z_2 - z_3}\right)$
Then:
== Sources ==
* : $1$: Complex Numbers: Solved Problems: Miscellaneous Problems: $51$
Category:Vertices of Equilateral Triangle in Complex Plane
Vieta's Formula for Pi

== Theorem ==
:$\pi = 2 \times \dfrac 2 {\sqrt 2} \times \dfrac 2 {\sqrt {2 + \sqrt 2} } \times \dfrac 2 {\sqrt {2 + \sqrt {2 + \sqrt 2} } } \times \dfrac 2 {\sqrt {2 + \sqrt {2 + \sqrt {2 + \sqrt 2 } } } } \times \cdots$
== Proof ==
Thus:
Then we have from the Half Angle Formula for Cosine that:
So we can replace all the instances of $\cos \dfrac \pi 4$, $\cos \dfrac \pi 8$, etc. with their expansions in square roots of $2$.
Finally, we note that from Limit of $\dfrac {\sin x} x$ at Zero we have:
:$\ds \lim_{\theta \mathop \to 0} \frac {\sin \theta} \theta = 1$
As $n \to \infty$, then, we have that $\dfrac \pi {2^n} \to 0$, and so:
:$\ds \lim_{n \mathop \to \infty} \frac {\map \sin {\pi / 2^n} } {\pi / 2^n} = 1$
The result follows after some algebra.
== Also known as ==
== Historical Note ==
== Sources ==
* : $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$
* : Chapter $\text {B}.9$: A proof of Vieta's Formula
* : $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$
* 
Category:Vieta's Formula for Pi
Category:Formulas for Pi
Category:Examples of Infinite Products
Vieta's Formula for Pi/Also known as

== Vieta's Formula for Pi: Also known as == 
Vieta's formula for pi is also known as Vieta's product, or Viète's product.
Some sources refer to it merely as Vieta's formula, or Viète's formula, but this can be confused with Viète's formulas in the context of polynomial theory.
The form Viète's formula for pi can of course also be seen.
== Sources ==
* 
* : Chapter $\text {B}.9$: A proof of Vieta's Formula
Category:Vieta's Formula for Pi
Vieta's Formula for Pi/Historical Note

== Historical Note on Vieta's Formula for Pi ==
, or as he is otherwise known, published his formula for $\pi$ in $1592$.
Some sources suggest that it was $1593$.
== Sources ==
* : $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$
* : Chapter $\text {B}.9$: A proof of Vieta's Formula
* : $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$
* 
Category:Historical Notes
Vinogradov's Theorem

== Theorem ==
Let $\Lambda$ be the von Mangoldt function.
For $N \in \Z$, let:
:$\ds \map R N = \sum_{n_1 + n_2 + n_3 \mathop = N} \map \Lambda {n_1} \, \map \Lambda {n_2} \, \map \Lambda {n_3}$
be a weighted count of the number of representations of $N$ as a sum of three prime powers.
Let $\SS$ be the arithmetic function:
:$\ds \map \SS N = \prod_{p \mathop \nmid N} \paren {1 + \frac 1 {\paren {p - 1}^3} } \prod_{p \mathop \divides N} \paren {1 - \frac 1 {\paren {p - 1}^2} }$
where:
:$p$ ranges over the primes
:$p \nmid N$ denotes that $p$ is not a divisor of $N$
:$p \divides N$ denotes that $p$ is a divisor of $N$.
Then for any $A > 0$ and sufficiently large odd integers $N$:
:$\map R N = \dfrac 1 2 \map \SS N N^2 + \map \OO {\dfrac {N^2} {\paren {\log N}^A} }$
where $\OO$ denotes big-$\OO$ notation.
=== Corollary 1 ===
=== Corollary 2 ===
== Proof ==
Throughout the proof, for $\alpha \in \R$, let the following notation be understood:
:$\map e \alpha := \map \exp {2 \pi i \alpha}$
Let $B > 0$, and set $Q = \paren {\log N}^B$.
For $1 \le q \le Q, 0 \le a \le q$ such that $\gcd \set {a, q} = 1$, let:
:$\map \MM {q, a} := \set {\alpha \in \closedint 0 1: \size {\alpha - \dfrac a q} \le \dfrac Q N}$
Let:
:$\ds \MM := \bigcup {1 \mathop \le q \mathop \le Q} \bigcup_{\substack {0 \mathop \le a \mathop \le q} {\gcd \set {a, q} \mathop = 1} } \map \MM {q, a}$
be referred to as the major arcs.
Let:
:$m := \closedint 0 1 \setminus \MM$
be referred to as the minor arcs.
=== Lemma 1 ===
By the Vinogradov Circle Method (with $\ell = 3$ and $\AA$ the set of primes), letting $\ds \map F \alpha = \sum_{n \mathop \le N} \map \Lambda n \, \map e {\alpha n}$ we have:
:$\ds \map R N = \int_0^1 \map F \alpha^3 \, \map e {-N \alpha} \rd \alpha$
So by splitting the closed unit interval into a disjoint union:
:$\closedint 0 1 = m \cup \MM$
we have:
:$\ds \map R N = \int_m \map F \alpha^3 \, \map e {-\alpha N} \rd \alpha + \int_\MM \map F \alpha^3 \, \map e {-\alpha N} \rd \alpha$
We consider each of these integrals in turn.
=== Sum Over the Minor Arcs ===
=== Sum Over the Major Arcs ===
Putting these estimates together, we obtain:
:$\map R N = \dfrac {N^2} 2 \map \SS N + \map \OO {\dfrac {N^2} {\paren {\log N}^{B / 2 - 5} } } + \map \OO {\dfrac {N^2} {\paren {\log N}^{B/2} } }$
Now choose $B$ carefully.
== Historical Note ==
== Sources ==
* 
* 
Category:Vinogradov's Theorem
Category:Analytic Number Theory
Category:Number Theory
Vinogradov's Theorem/Corollary 1

== Corollary to Vinogradov's Theorem ==
Let $\SS$ be the arithmetic function:
:$\ds \map \SS N = \prod_{p \mathop \nmid N} \paren {1 + \frac 1 {\paren {p - 1}^3} } \prod_{p \mathop \divides N} \paren {1 - \frac 1 {\paren {p - 1}^2} }$
where:
:$p$ ranges over the primes
:$p \nmid N$ denotes that $p$ is not a divisor of $N$
:$p \divides N$ denotes that $p$ is a divisor of $N$.
Let:
:$\ds \map r N = \sum_{p_1 + p_2 + p_3 \mathop = N} 1$
where $p_1, p_2, p_3$ are prime.
Then for sufficiently large odd integers $N$:
:$\map r N = \map \SS N \dfrac {N^2} {2 \paren {\log N}^3} \paren {1 + \map \OO {\dfrac {\log \log N} {\log N} } }$
where $\OO$ denotes big-O notation.
== Proof ==
Category:Number Theory
Category:Vinogradov's Theorem
Vinogradov's Theorem/Historical Note

== Historical Note on Vinogradov's Theorem ==
 was conjectured by in $1770$ in .
 proved what is now known as Vinogradov's theorem in $1937$, and as a result proved that $2$nd corollary: that every sufficiently large odd integer is the sum of three prime numbers.
This was considered an important step towards the resolution of Goldbach's Conjecture.
== Sources ==
* 
* 
* 
* 
Category:Historical Notes
Vinogradov's Theorem/Lemma 1

== Lemma ==
For sufficiently large $N$ the major arcs are pairwise disjoint, and the minor arcs are non-empty.
== Proof ==
Suppose that for some admissible $\dfrac {a_1} {q_1} \ne \dfrac {a_2} {q_2}$ we have:
:$\map \MM {q_1, a_1} \cap \map \MM {q_2, a_2} \ne \O$
Then using the definition of the major arcs, for $\alpha$ in the intersection we have:
and
This shows that:
:$N \le 2 Q^3 = 2 \paren {\log N}^3$
But by Power Dominates Logarithm, this is not the case for sufficiently large $N$.
Therefore the major arcs must be disjoint.
We have that the major arcs are pairwise disjoint closed intervals.
So by Cover of Interval By Closed Intervals is not Pairwise Disjoint it is not possible that $\MM = \closedint 0 1$.
So it follows that:
:$\MM \ne \O$
Category:Vinogradov's Theorem
Vinogradov's Theorem/Major Arcs

== Theorem ==
Let $B \in \R_{>0}$.
Then:
:$\ds \int_\MM \map F \alpha^3 \map e {-N \alpha} \rd \alpha = \frac {N^2} 2 \map \SS N + \map \OO {\frac {N^2} {\paren {\ln N}^{B/2} } }$
where the implied constant depends only on $B$.
== Proof ==
=== Lemma 1 ===
=== Lemma 2 ===
Let $N \in \N_{\ge 1}$.
Let $\beta \in \R$.
Let:
:$\ds \map u \beta = \sum_{n \mathop \le N} \map e {n \beta}$
For $P \ge 1$, define:
:$\ds \map {J_P} N = \int_{-P/N}^{P/N} \map u \beta^3 \map e {-N \beta} \rd \beta$
:$\map J N = \map {J_{N/2} } N$
Then with $Q = \paren {\ln N}^B$ as above:
:$\map {J_Q} N = \map J N + \map \OO {\dfrac {N^2} {Q^2} }$
and:
:$\map J N = \dfrac {N^2} 2 + \map \OO N$
=== Lemma 3 ===
Let $\alpha \in \map \MM {q, a}$ for some $q, a$ such that:
:$\map \MM {q, a} \subseteq \MM$
Let $\beta = \alpha - \dfrac a q$.
Then:
:$\map F \alpha^3 = \dfrac {\map \mu q} {\map \phi q^3} \map u \beta^3 + \map \OO {N^3 \map \exp {-C \sqrt {\ln N} } }$
where $C$ is a constant that depends only on $B$.
Category:Vinogradov's Theorem
Vinogradov's Theorem/Major Arcs/Lemma 1

== Lemma ==
Let $\phi$ be the Euler $\phi$ function.
Let $\mu$ be the Möbius function.
Let $c_q$ be the Ramanujan sum modulo $q$.
Let $P, N \ge 1$.
Let:
:$\ds \map {\SS_P} N := \sum_{q \mathop \le P} \frac {\map \mu q \map {c_q} N} {\map \phi q^3}$
:$\ds \map \SS N := \lim_{P \mathop \to \infty} \map {\SS_P} N$
Then:
:$\map \SS N = \map {\SS_P} N + \map \OO {P^{\epsilon -1} }$
and $\SS$ has the Euler product:
:$\ds \map \SS N = \prod_{p \mathop \nmid N} \paren {1 + \frac 1 {\paren {p - 1}^3} } \prod_{p \mathop \divides N} \paren {1 - \frac 1 {\paren {p - 1}^2} }$
where:
:$p \nmid N$ denotes that $p$ is not a divisor of $N$
:$p \divides N$ denotes that $p$ is a divisor of $N$.
== Proof ==
We have:
Trivially we have $\size {\map \mu q} \le 1$.
Therefore:
Therefore by Convergence of Powers of Reciprocals:
:$\map \SS N = \map {\SS_P} N + \map \OO {P^{\epsilon - 1} }$
as claimed.
By definition of Euler product, and because $\map \mu {p^k} = 0$ for $k > 1$:
:$\ds \map \SS N = \prod_p \paren {1 + \frac {\map \mu q \map {c_q} N} {\map \phi q^3} }$
Now for a prime $p$ we have:
:$\map \mu p = -1$
:$\map \phi p = p - 1$
We also have Kluyver's Formula for Ramanujan's Sum:
:$\ds \map {c_p} n = \sum_{d \mathop \divides \map \gcd {p, n} } \rd \map \mu {\frac p d}$
Let $p \divides N$.
Then:
$\map \gcd {p, N} = p$
which gives:
:$\map {c_p} N = p \map \mu 1 + \map \mu p = p - 1$
Let $p \nmid N$.
Then:
:$\map \gcd {p, N} = 1$
and so:
:$\map {c_p} N = -1$
Therefore:
:$\ds \map \SS N = \prod_{p \mathop \divides N} \paren {1 - \frac 1 {\paren {p - 1}^2} } \prod_{p \mathop \nmid N} \paren {1 + \frac 1 {\paren {p - 1}^3} }$
This completes the proof.
Category:Vinogradov's Theorem
Vinogradov's Theorem/Minor Arcs

== Theorem ==
Let $\ds \map F \alpha = \sum_{n \mathop \le N} \map \Lambda n \map e {\alpha n}$.
For any $B > 0$:
:$\ds \int_\MM \map F \alpha^3 \map e {-\alpha N} \rd \alpha \ll \frac {N^2} {\paren {\ln N}^{B/2 - 5} }$
== Proof ==
=== Lemma 1 ===
=== Lemma 2 ===
Let $a, q \in \Z$ such that:
:$(1): \quad \ds \size {\alpha - \frac a q} \le \frac 1 {q^2} \quad 1 \le q \le N, \quad \gcd \set {a, q} = 1$
Let $m, n \in \Z$ be integers.
Then:
:$\ds \sum_{k \mathop = 1}^m \min \set {\frac {m n} k, \frac 1 {\norm {\alpha k} } } \ll \paren {m + \frac {m n} q + q} \map \ln {2 q m}$
=== Lemma 3 ===
Let $\alpha$ satisfy the condition $(1)$ of Lemma 2.
Let $x, y \in \N$.
Let $\beta_k, \gamma_k$ be any complex numbers such that $\size {\beta_k}, \size {\gamma_k} \le 1$.
Then:
:$\ds \sum_{k \mathop = y}^{x/y} \sum_{\ell \mathop = y}^{x/k} \alpha_k \beta_\ell \map e {\alpha k \ell} \ll x^{1/2} \paren {\ln x}^2 \paren {\frac x y + y + \frac x q + q}^{1/2}$
Category:Vinogradov's Theorem
Vinogradov's Theorem/Minor Arcs/Lemma 1

== Lemma ==
For $\beta \in \R$, define:
:$\norm \beta := \min \set {\size {n - \beta}: n \in \Z}$
Then:
:$\ds \forall \alpha \in \R: \size {\sum_{k \mathop = N_1}^{N_2} \map e {\alpha k} } \le \min \set {N_2 - N_1, \frac 1 {2 \norm \alpha} }$
== Proof ==
The bound $N_2 - N_1$ is trivial:
Since $\size {\map e {\alpha k} } = 1$ for all $k$, by the Triangle Inequality:
:$\ds \size {\sum_{k \mathop = N_1}^{N_2} \map e {\alpha k} } \le \sum_{k \mathop = N_1}^{N_2} 1 = N_2 - N_1$
To show the second bound we evaluate the sum as a geometric series.
We have:
:$\map e {\alpha k} = \map e \alpha^k$
So by Sum of Geometric Sequence:
By the polar form of a complex number:
:$\map e \alpha - 1 = \map e {\alpha / 2} \paren {\map e {\alpha / 2} - \map e {-\alpha / 2} }$
and:
:$\map e {\alpha / 2} - \map e {-\alpha / 2} = \map \exp {\pi i \alpha} - \map \exp {-\pi i \alpha} = 2 i \map \sin {\pi \alpha}$
Therefore:
:$\ds \size {\sum_{k \mathop = N_1}^{N_2} \map e {\alpha k} } \le \frac 1 {\size {\map \sin {\pi \alpha} } }$
We know that the sine function is concave on $\closedint 0 {\pi / 2}$.
So:
:$\map \sin {\pi \alpha} \ge 2 \alpha$
for $\alpha \in \closedint 0 {1/2}$.
By definition there exists $n \in \Z$ such that:
:$\alpha = n + \norm \alpha$
and:
:$\norm \alpha \in \closedint 0 {\pi/2}$.
So:
This completes the proof.
Category:Vinogradov's Theorem
Vinogradov Circle Method

== Theorem ==
Let $\AA$ be a subset of the non-negative integers.
For $\alpha \in \R$, let:
:$\map e \alpha := \map \exp {2 \pi i \alpha}$
Let:
:$\ds \map {T_N} s = \sum_{\substack {a \mathop \in \AA \\ a \mathop \le N} } s^a$
be the truncated generating function for $\AA$.
Let:
:$\map {V_N} \alpha := \map {T_N} {\map e \alpha}$
Let $\map {r_{\AA, \ell} } N$ be the number of solutions $\tuple {x_1, \dotsc, x_\ell} \in \AA^\ell$ to the equation:
:$(1): \quad x_1 + \cdots + x_l = N$
Then:
:$\ds \map {r_{\AA, \ell} } N = \int_0^1 \map {V_N} \alpha^\ell \map e {-N \alpha} \rd \alpha$
== Proof ==
For $m \in \N$ let $\map {r_{\AA, \ell} } {m; N}$ be the number of solutions to $(1)$ with no $x_i$ exceeding $N$.
Then:
:$\forall m \le N: \map {r_{\AA, \ell} } {m; N} = \map {r_{\AA, \ell} } m$
and:
:$\forall m > \ell N: \map {r_{\AA, \ell} } {m; N} = 0$
Then we compute:
:$\ds \map {T_N} s^\ell = \sum_{m \mathop = 0}^{\ell N} \map {r_{\AA, \ell} } {m; N} s^m$
and:
:$(2): \quad \ds \map {V_N} \alpha^\ell = \sum_{m \mathop = 0}^{\ell N} \map {r_{\AA, \ell} } {m; N} \map e {\alpha m}$
Now it follows from Exponentials Form Orthonormal Basis for $\LL^2$ that:
:$\ds \int_0^1 \map e {\alpha m} \map e {-\alpha n} \rd \alpha = \delta_{m n}$
where $\delta_{m n}$ is the Kronecker delta.
Therefore, we multiply $(2)$ by $\map e {-N \alpha}$ and integrate:
:$\ds \map {r_{\AA, \ell} } N = \map {r_{\AA, \ell} } {N; N} = \int_0^1 \map {V_N} \alpha^\ell \map e {-N \alpha} \rd \alpha$
Category:Analytic Number Theory
Vitali's Convergence Theorem

== Theorem ==
Let $U$ be an open, connected subset of $\C$.
Let $S \subseteq U$ contain a limit point $\sigma$.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a normal family of holomorphic mappings $f_n : U \to \C$.
Let $\sequence {f_n}_{n \mathop \in \N}$ converge to some holomorphic mapping $f : U \to \C$ at $\sigma$.
Then $f_n$ converges uniformly to $f$ on all compact subsets of $U$.
== Proof ==
 there exists some compact subset $K$ of $U$ such that $f_n$ does not converge uniformly to $f$ on $K$.
Consider $K^* := K \cup \set \sigma$.
From Subsets Inherit Uniform Convergence, $f_n$ does not converge uniformly to $f$ on $K^*$.
From Uniformly Convergent iff Difference Under Supremum Norm Vanishes, the above is equivalent to:
:$\exists \epsilon > 0 : \forall N \in \N : \exists n \ge N : \norm {f_n - f}_{K^*} \ge \epsilon$
where $\norm {\cdot}_{K^*}$ denotes the supremum norm over $K^*$.
From Finite Union of Compact Sets is Compact, $K^{*}$ is compact.
Since $\sequence {f_n}$ is a normal family, there is some subsequence $\sequence {f_{n_r} }$ of $\sequence {f_n}$ and some mapping $g \in \map {\HH} U$ such that:
:$\sequence {f_{n_r} }$ converges uniformly to $g$ on $K^*$.
Further:
From the Identity Theorem, $f$ and $g$ agree on $U$.
From Uniformly Convergent iff Difference Under Supremum Norm Vanishes:
:$\exists N \in \N: r \ge N \implies \norm {f_{n_r} - f}_{K^*} < \epsilon$
This contradicts the result that:
:$\forall N \in \N: \exists n \ge N: \norm {f_n - f}_{K^*} \ge \epsilon$
Hence the result, by Proof by Contradiction.
== Sources ==
* : $\S 5.2.1$
Category:Complex Analysis
Vitali's Theorem

== Theorem ==
Let $\struct {X, \Sigma, \mu}$ be a $\sigma$-finite measure space, and let $p \in \R, p \ge 1$.
Let $\sequence {f_n}_{n \mathop \in \N}: f_n: X \to \R$ be a sequence of $p$-integrable functions.
Also, let $f: X \to \R$ be a measurable function.
Suppose that $\ds \operatorname {\mu-\!\lim\,} \limits_{n \mathop \to \infty} f_n = f$, that is $f_n$ converges in measure to $f$.
Then the following are equivalent:
:$(1): \quad \ds \lim_{n \mathop \to \infty} \norm {f_n - f}_p = 0$, where $\norm {\,\cdot\,}_p$ is the $p$-seminorm (that is $\ds \operatorname {\LL^{\textit p}-\!\lim\,} \limits_{n \mathop \to \infty} f_n = f$)
:$(2): \quad \sequence {\size {f_n}^p}_{n \mathop \in \N}$ is a uniformly integrable collection
:$(3): \quad \ds \lim_{n \mathop \to \infty} \int \size {f_n}^p \rd \mu = \int \size f^p \rd \mu$
If $\struct {X, \Sigma, \mu}$ is not $\sigma$-finite, $(1)$ and $(3)$ must be replaced by, respectively:
:$(1'): \quad \sequence {f_n}_{n \mathop \in \N}$ converges in $\LL^p$
:$(3'): \quad \ds \lim_{n \mathop \to \infty} \int \size {f_n}^p \rd \mu$ exists in $\R$
== Proof ==
== Sources ==
* : $16.6, 16.7$
Category:Measure Theory
Vitali-Carathéodory Theorem

== Theorem ==
Let $\struct {X, \tau}$ be a locally compact Hausdorff space.
Let $\MM$ be a $\sigma$-algebra over $X$ which contains the Borel $\sigma$-algebra generated by $\tau$.
Let $\mu$ be a Radon measure on $\MM$.
Let $f \in \map {\LL^1} \mu$, where $\map {\LL^1} \mu$ denotes the (real) Lebesgue 1-space of $\mu$.
For all $\epsilon \in \R_{>0}$, there exists some $\tuple {u, v} \in \paren {X^\R}^2$ such that:
:$u$ is upper semicontinuous and bounded above
:$v$ is lower semicontinuous and bounded below
:$u \le f \le v$
and :
:$\ds \int_X \paren {v - u} \rd \mu < \epsilon$.
== Proof ==
Let:
:$\forall x \in X: \map f x \ge 0$
and:
:$\exists x \in X: \map f x \ne 0$
By Measurable Function is Pointwise Limit of Simple Functions, there exists a sequence:
:$\sequence {s_n} \in \paren {\map \EE \MM}^\N$
where $\map \EE \MM$ denotes the space of simple functions on $\struct {X, \MM}$.
By Pointwise Difference of Simple Functions is Simple Function, the differences of consecutive terms in a sequence of simple functions are simple functions.
By Limit of Sequence is Sum of Difference of Consecutive Terms, there exists a sequence $\sequence {t_n}$ of simple functions such that:
:$\ds f = \sum_{i \mathop = 1}^\infty t_n$
By the definition of simple functions, each simple function is a finite linear combination of characteristic functions.
Thus there exists some $\tuple {\sequence {E_i}, \sequence {c_i} } \in \MM^\N \times \R_{>0}^\N$ such that:
:$\ds f = \sum_{i \mathop = 1}^\infty c_i \chi_{E_i}$
Now:
That is, the series:
:$\ds \sum_{i \mathop = 1}^\infty c_i \map \mu {E_i}$
converges. 
Denote by $\powerset X$ the power set of X.
By the definition of Radon measure, for all $\epsilon \in \R_{>0}$, there exists some $\tuple {\sequence {K_i}, \sequence {V_i} } \in \paren {\paren {\powerset X}^\N}^2$ such that for all $i \in \N$:
:$K_i$ is compact
:$V_i$ is open
:$K_i \subset E_i\subset V_i$
and
:$c_i \map \mu {V_i - K_i} < 2^{-\paren {i + 1} } \epsilon$
By Characteristic Function of Open Set is Lower Semicontinuous:
:for all $i \in \N$, $\chi_{V_i}$ is lower semicontinuous. 
By Constant Multiple of Lower Semicontinuous Function is Lower Semicontinuous:
:for all $i \in \N$, $c_i \chi_{V_i}$ is lower semicontinuous.
Define:
:$\ds v = \sum_{i \mathop = 1}^\infty c_i \chi_{V_i}$
By Series of Lower Semicontinuous Functions is Lower Semicontinuous:
:$v$ is lower semicontinuous.
By Characteristic Function of Compact Set is Upper Semicontinuous:
:for all $i \in \N$, $\chi_{K_i}$ is upper semicontinuous.
By Constant Multiple of Upper Semicontinuous Function is Upper Semicontinuous:
:for all $i \in \N$, $c_i \chi_{K_i}$ is upper semicontinuous.
By definition of convergent series, for all there exists some $N \in \N$ such that:
:$\ds \sum_{i \mathop = N + 1}^\infty c_i \map \mu {E_i} < \frac \epsilon 2$
Define:
:$\ds u = \sum_{i \mathop = 1}^N c_i \chi_{K_i}$
By Finite Sum of Upper Semicontinuous Functions is Upper Semicontinuous, $u$ is upper semicontinuous.
Now, for all $i \in \N$:
:$\chi_{K_i} \le \chi_{E_i} \le \chi_{V_i}$
So:
:$u \le f \le v$
Now:
Category:Real Analysis
Category:Hausdorff Spaces
Vitali Set Existence Theorem

== Theorem ==
There exists a set of real numbers which is not Lebesgue measurable.
== Proof 1 ==
== Proof 2 ==
== Also presented as ==
== Also known as ==
== Also see ==
The set $M$ defined here is an example of a Vitali set: a subset of the real numbers which has no Lebesgue measure.
== Linguistic Note ==
== Sources ==
* 
* : $1.$ Introduction: $1.2$ A nonmeasurable set of real numbers
* 
* 
Category:Vitali Set Existence Theorem
Category:Vitali Sets
Category:Non-Measurable Sets
Category:Measure Theory
Vitali Set Existence Theorem/Also presented as

== Theorem ==
== Proof ==
The proof of the Vitali Set Existence Theorem is presented in using the following argument:
 $M$ were Lebesgue measurable.
First, $\map \mu M = 0$ is impossible, because from $(1)$ this would mean $\map \mu \R = 0$.
Suppose $\map \mu M > 0$.
Then:
So $\map \mu M > 0$ is also impossible.
Hence $M$ is not Lebesgue measurable.
It needs to be pointed out that this:
:$\ds \map \mu {\closedint 0 1} \ge \map \mu {\bigcup \set {M_r: r \in \Q \land 0 \le r \le 1} }$
is not immediately obvious, as it is not the case that:
:$\ds \closedint 0 1 \supseteq \bigcup \set {M_r: r \in \Q \land 0 \le r \le 1}$
However, because $\map \mu X$ is translation invariant, $M_r$ has the same measure as $M_0$.
Hence the result.
== Sources ==
* : $1.$ Introduction: $1.2$ A nonmeasurable set of real numbers
* 
* 
Category:Vitali Set Existence Theorem
Vitali Set Existence Theorem/Lemma

== Theorem ==
For all real numbers in the closed unit interval $\mathbb I = \closedint 0 1$, define the relation $\sim$ such that:
:$\forall x, y \in \mathbb I: x \sim y \iff x - y \in \Q$
where $\Q$ is the set of rational numbers.
That is, $x \sim y$ their difference is rational.
Then $\sim$ is an equivalence relation.
== Proof ==
Checking in turn each of the criteria for equivalence:
=== Reflexivity ===
:$\forall x \in \mathbb I: x - x = 0 \in \Q$
Thus $\sim$ is reflexive.
=== Symmetry ===
By Inverse for Rational Addition:
:$\forall x, y \in \mathbb I: x - y \in \Q \implies y - x = -\paren {x - y} \in \Q$
Thus $\sim$ is symmetric.
=== Transitivity ===
Let $x, y, z \in \mathbb I$.
Suppose $x - y \in \Q$ and $y - z \in \Q$.
Then by Rational Addition is Closed:
:$x - z = \paren {x - y} + \paren {y - z} \in \Q$
Thus $\sim$ is transitive.
Thus we have shown that $\sim$ is an equivalence relation.
Category:Vitali Set Existence Theorem
Vitali Set Existence Theorem/Linguistic Note

== Linguistic Note on Vitali Set Existence Theorem ==
Some sources refer to this as the Vitali Theorem, but this can easily be confused with the result known on as Vitali's Theorem, which is a completely different result.
However, as both results are in the field of measure theory, it is not possible to distinguish them by context.
Many sources quote and analyse the result without giving it a name.
Category:Vitali Set Existence Theorem