Articles of measure theory

Can $\int|f_n|d\mu \to \int |f|d\mu$ but not $\int|f_n – f|d\mu \to 0$?

Possible Duplicate: Convergence a.e. and of norms implies that in Lebesgue space I am trying to show that if $$ \int_X |f_n|d\mu \to \int_X|f|d\mu $$ where $f$ and all the $f_n$ have finite integral and $f_n \to f$ pointwise, then $$ \int_X |f_n-f|d\mu \to 0. $$ I worked out a proof in the case that […]

Almost Everywhere Convergence versus Convergence in Measure

I am having some conceptual difficulties with almost everywhere (a.e.) convergence versus convergence in measure. Let $f_{n} : X \to Y$. In my mind, a sequence of measurable functions $\{ f_{n} \}$ converges a.e. to the function $f$ if $f_{n} \to f$ everywhere on $X$ except for maybe on some set of measure zero. However, […]

Limit of measures is again a measure

Given a sequence $(\mu_n)_{n\in \mathbb N}$ of finite measures on the measurable space $(\Omega, \mathcal A)$ such that for every $A \in \mathcal A$ the limit $$\mu(A) = \lim_{n\to \infty} \mu_n(A)$$ exists. I want to show that $\mu$ is a measure on $\mathcal A$. What I managed to figure out: $\mu$ is monotone, additive and […]

Is composition of measurable functions measurable?

We know that if $ f: E \to \mathbb{R} $ is a Lebesgue-measurable function and $ g: \mathbb{R} \to \mathbb{R} $ is a continuous function, then $ g \circ f $ is Lebesgue-measurable. Can one replace the continuous function $ g $ by a Lebesgue-measurable function without affecting the validity of the previous result?

Measurability of one Random Variable with respect to Another

After several hours of struggling, I’ve been unable to solve the following problem Let $X,Y: (\Omega, \mathcal{S}) \rightarrow (\mathbb{R}, \mathcal{R})$ where $\mathcal{R}$ are the Borel Sets for the Reals. Show that $Y$ is measurable with respect to $\sigma(X) = \{ X^{-1}(B) : B \in \mathcal{R} \}$ if and only if there exists a function $f: […]

$L_{p}$ distance between a function and its translation

I’m working through a proof and one of the comments is that for a function $f\in L_p (\mathbb{T})$: $$\lim_{t\to 0}\;\|f(\cdot + t) – f\|_p = 0.$$ How do I prove it? I think it is intuitively clear if $f$ is a step function, but what about for an arbitrary $p$ integrable function?

Singular continuous functions

A function $f:[0,1]\rightarrow\mathbb{R}$ is called singular continuous, if it is nonconstant, nondecreasing, continuous and $f^\prime(t)=0$ whereever the derivative exists. Let $f$ be a singular continuous function and $T$ the set where $f$ is not differentiable. Question: Is $T$ nowhere dense? Examples: A classical example of such a function is the so-called devil’s staircase, obtained as […]

Weird measurable set

In the following, consider the Lebegue measure in $\mathbb{R}^d$. Consider $E\subseteq \mathbb{R}^d$ measurable, with $0\lt m(E)\lt\infty$, such that any measurable subset $F$ of $E$ satisfies $m(F)=m(E)$ or $m(F)=0$. What can we say about $E$? Does there exist such a set?

How to find an irrational number in this case?

From Baire category theorem, we see that $\mathbb{Q}$ can not be a $G_{\delta}$. But consider the following construction: Let us consider $\mathbb{Q}\cap [0,1]$, putting all the elements in the set in a sequence, denoted $\{a_n\}$. We define $$V_i=\bigcup_{j}[a_j-1/2^{i+j},a_j+1/2^{i+j}]\cap [0,1].$$ Notice that $\mathbb{Q}\subset V_i$. So we define $$V=\bigcap_{i} V_i.$$ We have $\mathbb{Q}\subset V$, and $V$ is […]

How can this theorem about weakly measurable functions on $\sigma$-finite measure spaces be deduced from the finite measure space case?

I am reading a theorem about measurability of vector-valued functions in a note on functional analysis: Theorem 3.6.1. If $X$ is a separable, metrizable locally convex space, $(\Omega, \Sigma, \mu)$ is a $\sigma$-finite measure space, and $f : \Omega \to X$ is weakly measurable, then $f$ is strongly measurable. The proof begins We can restrict […]