$f,g \in [0,1] \times [0,1]$, $\int f – g \mathrm{d}x = 0$ and are monotonically increasing, then $\int |f-g| \mathrm{d}x \le \frac{1}{2}$

$ f,g$ are monotonically increasing in $ [0,1]$ and $ 0\le f , g \le 1$ . $ \int_0^1 f – g \mathrm{d}x = 0$ . Prove that

$ $ \int_0^1 |f – g|\mathrm{d}x \le \frac{1}{2}$ $

In my previous question, $ g(x) = x$ . And my teacher said $ x$ can be replaced by $ g(x)$ . In fact, in previous question, we don’t need to use the condition $ \int_0^1 f – g \mathrm{d}x = 0 $ . But if we replace $ x$ with $ g$ , this condition becomes necessary.

Also, if $ g = x$ , we can replace the $ \frac{1}{2}$ with $ \frac{1}{4}$ ,that is

$ $ \int_0^1|f-g| \mathrm{d}x \le \frac{1}{4}$ $ I am wondering how to prove that.

show that $\int \int_{S}^{}{curl\vec{F}\cdot \vec{dS}}$ is proportional to the lenght of $C$

The curve $ C$ is the edge of a surface S , with $ \vec{T}$ unit tangent vector of the curve $ C$ and $ \vec{F}$ a vector field such as $ \vec{F}=k\vec{T} $ for each point of $ C$ , where $ k$ is a constant , how can I prove that

$ \int \int_{S}^{}{curl\vec{F}\cdot \vec{dS}}\; $ is proportional to the length of $ C$

Any help would be appreciate . Thanks in advance.

If $f$ belongs to $M^{+} $ and $c \ge 0$ then $cf$ belongs to $M^{+}$ and $ \int cf = c\int f$

If $ f$ belongs to $ M^{+} $ and $ c \ge 0$ then $ cf$ belongs to $ M^{+}$ and $ \int cf = c\int f$ .

I need to proove that, using the following observation:

if $ f\in M^{+}$ and $ c>0 $ , then the mapping $ \varphi \rightarrow \psi = c\varphi$ is a one-toone mapping between simple function $ \varphi \in M^{+}$ with $ \varphi \le f $ and simple functions $ \varphi$ in $ M^{+} $ with $ \psi \le cf $ .

I know that this question is already answer here:One-to-one mapping of simple functions $ \phi \to \psi = c\,\phi$ implies $ \int cf\,d\mu = c \int f\,d\mu$ ?

But I can’t follow the verbal explanation.

My original idea was to proove $ $ c \int f \le \int cf \le c\int f $ $ But I can’t… some idea?

(local variable) int PlaceNumberValue Error: Cannot Implicity convert type ‘int’ to ‘string’

Buenas noches les dejo este pequeño error no puedo convertir la variable ‘PlaceNumberValue’ de string a entero aqui les dejo el codigo:

using System; using System.Collections.Generic; using System.Linq; using System.Text;

namespace TrackerLibrary { /// /// Represent what the prize if for the given place. /// public class PrizeModel { /// /// The unique identifier for the prize /// public int Id { get; set; } /// /// The numeric identifier for the place(for the second place, etc.) /// public int PlaceNumber { get; set; } /// /// The friendly name for the place (second place, first runner up, etc.) ///

    public string PlaceName { get; set; }      /// <summary>     /// The find amount this place earns or zero if it is not used.     /// </summary>      public decimal PrizeAmount { get; set; }     /// <summary>     /// The number that represent the percentage of theoverall take or     /// zero if it is not used. The percentage if a fraction of 1 (so O.S for     /// 50%)     /// </summary>      public double PrizePercentage { get; set; }      public PrizeModel()     {      }      public PrizeModel(string placeName, string placeNumber, string placeAmount, string placePercentage)     {         PlaceName = PlaceName;         int placeNumberValue = 0;         int.TryParse(placeNumber, out placeNumberValue);         placeNumber = placeNumberValue; //AQUI ESTA EL ERROR          decimal prizeAmountValue = 0;         decimal.TryParse(prizeAmount, out prizeAmountValue);         PrizeAmount = prizeAmountValue;     }       public string prizeAmount { get; set; } } 

}

Let $(X , \cal{A}, m)$ be a measure space. Let $f:X \to [0,1]$ be measurable. If $m(X) < \infty$, find$\lim_{n\to\infty} \int f^n \, d m$.

Let $ (X , \cal{A}, m)$ be a measure space. Let $ f:X \to [0,1]$ be a measurable function. If $ m(X) < \infty$ , determine $ \lim_{n\to\infty} \int f^n \, d m$ .

So far I have:

If $ f(x) < 1$ , then $ \lim_{n\to \infty}{f^n(x)} = 0$ . If $ f(x) = 1$ , then $ \lim_{n\to\infty}{f^n(x)} =1$ . So, for each $ x \in X$ , $ $ \lim_{n\to\infty}{f^n(x)} = \chi_{_{[f = 1]}}(x). $ $

However, I am stuck because I cannot use the Lebesgue Monotone Convergence Theorem, since the sequence is decreasing. Also, I do not know where I will use the hypothesis that $ X$ is a finite measure space. Any ideas?

Prove or disprove sentence about $\int f$

I post this question a few months ago. I solved the items (1) and (3), but I cannot to solve (2). Today I read this question again and I’m curious about solution of (2). Today, I had an idea, but I dont know if it works.


Idea. A monotone function has only jump discontinuities. So, $ f|_{[f(x_{0}^{-}),f(x_{0}^{+})]}$ is continuous, then there is a maximum and minimum. If $ w$ is a minimum on $ [f(x_{0}^{-}),f(x_{0}^{+})]$ , then

$ $ w \leq f(x) \Longrightarrow w(x-x_{0}) \leq \int_{x_{0}}^{x}f(t)dt = F(x) – F(x_{0})$ $

But, this works for $ f$ on $ [f(x_{0}^{-}),f(x_{0}^{+})]$ . What about the general case?

TypeError: unsupported operand type(s) for -: ‘int’ and ‘tuple’ – Subtração de dois elementos do vetor [pendente]

Olá, estou tentando subtrair dois elementos de duas listas diferentes, de acordo com as interações mas sempre está mostrando esse erro:

TypeError: unsupported operand type(s) for -: ‘int’ and ‘tuple’

Aqui está meu código:

 for i in range(0, Tam):      dist.append(sqrt(pow(R[i+1] - Rf[i], 2) + pow(G[i+1] - Gf[i], 2)))